Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring

The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Sriva...

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Datum:	2016
Hauptverfasser:	Karp, D., Prilepkina, E.
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Sprache:	English
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Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1477482019-02-16T01:25:29Z Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring Karp, D. Prilepkina, E. The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's G-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's G-function and generalized hypergeometric function around unity. Particular cases of these identities include known and new relations for Thomae's hypergeometric function and forgotten Hermite's identity for the sine function. 2016 Article Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring / D. Karp, E. Prilepkina // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C20; 33C60; 34M35 DOI:10.3842/SIGMA.2016.052 http://dspace.nbuv.gov.ua/handle/123456789/147748 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's G-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's G-function and generalized hypergeometric function around unity. Particular cases of these identities include known and new relations for Thomae's hypergeometric function and forgotten Hermite's identity for the sine function.
format	Article
author	Karp, D. Prilepkina, E.
spellingShingle	Karp, D. Prilepkina, E. Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Karp, D. Prilepkina, E.
author_sort	Karp, D.
title	Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring
title_short	Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring
title_full	Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring
title_fullStr	Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring
title_full_unstemmed	Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring
title_sort	hypergeometric differential equation and new identities for the coefficients of nørlund and bühring
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/147748
citation_txt	Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring / D. Karp, E. Prilepkina // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT karpd hypergeometricdifferentialequationandnewidentitiesforthecoefficientsofnørlundandbuhring AT prilepkinae hypergeometricdifferentialequationandnewidentitiesforthecoefficientsofnørlundandbuhring
first_indexed	2025-07-11T02:45:56Z
last_indexed	2025-07-11T02:45:56Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 052, 23 pages Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring? Dmitrii KARP †‡ and Elena PRILEPKINA †‡ † Far Eastern Federal University, 8 Sukhanova Str., Vladivostok, 690950, Russia ‡ Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, 7 Radio Str., Vladivostok, 690041, Russia E-mail: dimkrp@gmail.com, pril-elena@yandex.ru URL: http://dmkrp.wordpress.com Received February 25, 2016, in final form May 15, 2016; Published online May 21, 2016 http://dx.doi.org/10.3842/SIGMA.2016.052 Abstract. The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer’s G-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer’s G-function and generalized hypergeometric function around unity. Particular cases of these identities include known and new relations for Thomae’s hypergeometric function and forgotten Hermite’s identity for the sine function. Key words: generalized hypergeometric function; hypergeometric differential equation; Mei- jer’s G-function; Bernoulli polynomials; Nørlund’s coefficients; Bühring’s coefficients 2010 Mathematics Subject Classification: 33C20; 33C60; 34M35 1 Introduction We will use standard notation Z, N and C to denote integer, natural and complex numbers, respectively; N0 = N ∪ {0}. The hypergeometric differential equation (D = z d dz ) {(D − a1)(D − a2) · · · (D − ap)− z(D + 1− b1)(D + 1− b2) · · · (D + 1− bp)} y = 0 (1.1) for p = 2 was first considered by Euler and later studied by Gauss, Kummer, Riemann, Papperitz and Schwarz, among others, see [2, Section 2.3]. For general p > 2 it was probably first investigated by Thomae [42]. Note that our choice of parameters differs slightly from that of [3, (16.8.3)]. As will become apparent in the sequel, this choice is more convenient if the solution is to be built in terms of Meijer’s G-functions and not generalized hypergeometric functions. Equation (1.1) is of Fuchsian type and has three regular singularities located at the points 0, 1, ∞. The local exponents read [5, (2.6)–(2.8)] a1, a2, . . . , ap at z = 0, 0, 1, 2, . . . , p− 2, ∑ (bi − ai)− 1 at z = 1, 1− b1, 1− b2, . . . , 1− bp at z =∞. ?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica- tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html mailto:dimkrp@gmail.com mailto:pril-elena@yandex.ru http://dmkrp.wordpress.com http://dx.doi.org/10.3842/SIGMA.2016.052 http://www.emis.de/journals/SIGMA/OPSFA2015.html 2 D. Karp and E. Prilepkina The fundamental sets of solutions around the points z = 0 and z =∞ were found by Thomae in [42] and are expressed in terms of the generalized hypergeometric series pFq ( a b z ) = pFq (a;b; z) = ∞∑ n=0 (a1)n(a2)n · · · (ap)n (b1)n(b2)n · · · (bq)nn! zn, where (a)n = Γ(a + n)/Γ(a) denotes the rising factorial and a = (a1, . . . , ap), b = (b1, . . . , bq) are complex vectors such that −bi /∈ N0 for i = 1, . . . , q. If the components of a are distinct modulo Z, then the basis of solutions near z = 0 is given by [33, (1.13)] zakpFp−1 ( 1− b + ak 1− a[k] + ak z ) , k = 1, . . . , p, where a[k] signifies the vector a with the element ak omitted and a+α is understood as (a1 +α, . . . , ap + α). If the components of b are distinct modulo Z then the fundamental system of solutions near z =∞ is given by [33, (1.14)] zbk−1pFp−1 ( 1 + a− bk 1 + b[k] − bk 1 z ) , k = 1, . . . , p. Finally, the fundamental set of solutions around the point z = 1 was found by the remarkable Danish mathematician Niels Erik Nørlund in his milestone work [33]. These facts are well-known and have been frequently cited in the literature. It seems to be less known that the fundamental solutions constructed by Nørlund can be expressed in terms of G-function introduced some 15 years earlier by Meijer, see [30]. In fact, Meijer himself studied the hypergeometric differential equation more general than (1.1) and built the basis of solutions in the neighborhood of zero and infinity in terms of G-functions. The connection between Nørlund’s solutions and Meijer’s G-function was observed by Marichev and Kalla [27] and Marichev [26] but these two papers remained largely unnoticed. The fact that each solution around z = 1 must equal to a linear combination of fundamental solutions around z = 0 is reflected for p = 2 in the following identity due to Gauss [2, Theorem 2.3.2] 2F1 ( α1, α2 β ; 1− z ) = Γ(β)Γ(β − α1 − α2) Γ(β − α1)Γ(β − α2) 2F1 ( α1, α2 α1 + α2 − β + 1 ; z ) + Γ(β)Γ(α1 + α2 − β) Γ(α1)Γ(α2) zβ−α1−α2 2F1 ( β − α1, β − α2 β − α1 − α2 + 1 ; z ) . (1.2) The fact that each solution around z = 0 must equal to a linear combination of fundamental solutions around z = 1 leads to essentially the same identity with z replaced by 1 − z. For p > 2, however the above connection formula has two different generalizations. One of them is the expansion (2.4) of Gp,0p,p (a solution around z = 1) into a sum of pFp−1 (fundamental solutions around z = 0) which can also be obtained by applying the residue theorem to the definition of G-function and is found in standard references on G-function, see, for instance, [3, (16.17.2)]. The other one is the expansion of pFp−1 into a sum of Gp,0p,p with p − 1 instances of G2,p p,p as given by formula (2.3) below. This expansion, discovered by Nørlund without any mentioning of G-function has been reproduced in [27] in a slightly different form but seems to be forgotten afterwards. A related expansion, again with no reference to G-function, has been then found in [7, 8] by Bühring whose goal was to describe the behavior of pFp−1(z) near z = 1. The logarithmic cases have been studied by Saigo and Srivastava in [37]. Let us also mention that the monodromy group of the generalized hypergeometric differential equation has been constructed by Beukers and Heckman in [5]. Hypergeometric Differential Equation and New Identities for the Coefficients 3 This paper is organized as follows. Section 2 is of survey nature: we review the results of Nørlund and Bühring and rewrite them in terms of G-function. We also reveal connec- tions between Nørlund’s and Bühring’s expansions and relate them to various results obtained in statistics literature. In Section 3 we present some new formulas for Nørlund’s coefficients and utilize the above mentioned connections to derive various new identities for Nørlund’s and Bühring’s coefficients which reduce to hypergeometric and trigonometric identities for small p. The most striking of these identities, p∑ k=1 p∏ i=1 sin(βi − αk) p∏ i=1 i 6=k sin(αi − αk) = sin ( p∑ k=1 (βk − αk) ) , (1.3) can be viewed as a generalization of Ptolemy’s theorem: for a quadrilateral inscribed in a circle the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides. This theorem can be written in trigonometric form which yields precisely the above identity for p = 2. For p = 3 formula (1.3) was first presented without proof by Glaisher at a 1880 conference and then proved in full generality by Hermite in his 1885 paper [20]. The same paper contains a number of other amazing identities completely forgotten until the recent article [21] by Johnson, where many interesting mathematical and historical details can be found. Furthermore, if we substitute sin(z) by z in every occurrence of sin in (1.3), we obtain an identity discovered by Gosper, Ismail and Zhang in [19] and known as non-local derangement identity. It has been recently used by Feng, Kuznetsov and Yang to find new formulas for sums of products of generalized hypergeometric functions [18]. When this paper was nearly finished E. Scheidegger published a preprint [38] that also deals with the hypergeometric differential equation (1.1) and builds largely on the works of Nørlund [33] and Bühring [8]. Scheidegger suggests a new basis of solutions around 1 and presents its series expansion. Main emphasis in [38] is on the logarithmic cases (parameter differences are integers). Scheidegger’s work is motivated by applications in certain one-parameter families of Calabi–Yau manifolds, known as the mirror quartic and the mirror quintic. 2 Results of Nørlund and Bühring revisited 2.1 Fundamental solutions around unity The simple observation that nevertheless seems to be largely overlooked in the literature (except for [26, 27]) lies in the fact that Nørlund’s solutions to (1.1) can be expressed in terms of Meijer’s G-function. Let us remind its definition first. Suppose 0 ≤ m ≤ q, 0 ≤ n ≤ p are integers and a, b are arbitrary complex vectors, such that ai − bj /∈ N for all i = 1, . . . , n and j = 1, . . . ,m. Meijer’s G-function is defined by the Mellin–Barnes integral of the form (see [17, Section 5.3], [25, Chapter 1], [36, Section 8.2] or [3, Section 16.17]), Gm,np,q ( z a b ) := 1 2πi ∫ L Γ(b1 + s) · · ·Γ(bm + s)Γ(1− a1 − s) · · ·Γ(1− an − s)z−s Γ(an+1 + s) · · ·Γ(ap + s)Γ(1− bm+1 − s) · · ·Γ(1− bq − s) ds, (2.1) where the contour L is a simple loop that separates the poles of the integrand of the form bjl = −bj− l, l ∈ N0, leaving them on the left from the poles of the form aik = 1−ai+k, k ∈ N0, leaving them on the right [25, Section 1.1]. It may have one of the three forms L−, L+ or Liγ described below. Choose any ϕ1 < min{−=b1, . . . ,−=bm,=(1− a1), . . . ,=(1− an)}, ϕ2 > max{−=b1, . . . ,−=bm,=(1− a1), . . . ,=(1− an)} 4 D. Karp and E. Prilepkina and arbitrary real γ. The contour L− is a left loop lying in the horizontal strip ϕ1 ≤ =s ≤ ϕ2. It starts at the point −∞ + iϕ1, terminates at the point −∞ + iϕ2 and coincides with the boundary of the strip for sufficiently large \|s\|. Similarly, the contour L+ is a right loop lying in the same strip, starting at the point +∞+ iϕ1 and terminating at the point +∞+ iϕ2. Finally, the contour Liγ starts at γ − i∞, terminates at γ + i∞ and coincides with the line <s = γ for all sufficiently large \|s\|. The power function z−s is defined on the Riemann surface of the logarithm, so that z−s = exp(−s{log \|z\|+ i arg(z)}) and arg(z) is allowed to take any real value. Hence, Gm,np,q (z) is also defined on the Riemann surface of the logarithm. In order that the above definition be consistent one needs to prove that the value of the integral remains intact if convergence takes place for several different contours. Alternatively, one may split the parameter space into nonintersecting subsets and stipulate which contour should be used in each subset. Another key issue that must be ad- dressed with regard to the above definition is whether the integral in (2.1) equals the sum of residues of the integrand and, if yes, on which side of L the residues are to be counted. This is important for both theoretical considerations (expressing G-function in terms of hypergeometric functions) and especially for actually computing the value of G-function (although numerical contour integration can also be employed). In this paper we will only need the G-function of the form Gm,np,p . For this particular type of G-function the solutions to the above problems seem to be rather complete. We placed further details regarding the definition and answers to the above questions for Gm,np,p in Appendix A. A simple property of Meijer’s G-function implied by its definition (2.1) which will be fre- quently used without further mentioning is given by [36, (8.2.2.15)] zαGm,np,q ( z a b ) = Gm,np,q ( z a + α b + α ) , (2.2) where α ∈ C and a + α is understood as (a1 + α, . . . , ap + α). The functions Gp,0p,p and G2,p p,p will play a particularly important role in this paper, so that we found it useful to cite the known explicit expressions for small p. If p = 1 then [36, for- mula (8.4.2.3)] G1,0 1,1 ( z b a ) = za(1− z)b−a−1+ Γ(b− a) , where (x)+ = x for x ≥ 0 and 0 otherwise. If p = 2 we have [36, formula (8.4.49.22)] G2,0 2,2 ( z b1, b2 a1, a2 ) = za2(1− z)b1+b2−a1−a2−1+ Γ(b1 + b2 − a1 − a2) 2F1 ( b1 − a1, b2 − a1 b1 + b2 − a1 − a2 ; 1− z ) , and [36, formula (8.4.49.20)] G2,2 2,2 ( z b1, b2 a1, a2 ) = za1Γ(1 + a1 − b1)Γ(1 + a1 − b2)Γ(1 + a2 − b2)Γ(1 + a2 − b1) Γ(2 + a1 + a2 − b1 − b2) × 2F1 ( 1 + a1 − b1, 1 + a1 − b2 2 + a1 + a2 − b1 − b2 ; 1− z ) . If p = 3 we have [36, formula (8.4.51.2)] G3,0 3,3 ( z b1, b2, b3 a1, a2, a3 ) = za1+a2−b1−1(1− z)b1+b2+b3−a1−a2−a3−1+ Γ(b1 + b2 + b3 − a1 − a2 − a3) × F3(b1 − a2, b3 − a3; b1 − a1, b2 − a3; b1 + b2 + b3 − a1 − a2 − a3; 1− 1/z, 1− z), where F3 is Appell’s hypergeometric function of two variables [36, index of functions]. Hypergeometric Differential Equation and New Identities for the Coefficients 5 Having made these preparations we can formulate Nørlund’s result regarding a fundamental solution of (1.1) in the neighborhood of 1. First, introduce the following notation: ψm = m∑ i=1 (bi − ai), 1 ≤ m ≤ p, a = (a1, . . . , ap), b = (b1, . . . , bp), a[k] = (a1, . . . , ak−1, ak+1, . . . , ap), sin(a) = sin a1 sin a2 · · · sin ap, Γ(a) = Γ(a1)Γ(a2) · · ·Γ(ap) and let a[k,s] denote the vector a with the elements ak and as removed. We write <(a) > 0 for <(ai) > 0, i = 1, . . . , p. The next theorem is implicit in [33]. Theorem 2.1 (Nørlund). Suppose k, s ∈ {1, . . . , p} and us(z) = Gp,0p,p ( z b a ) , uk(z) = G2,p p,p ( z b ak, as,a[k,s] ) for k 6= s. Then the set {uk(z)}pk=1 forms the fundamental system of solutions of (1.1) in the neighborhood of z = 1 and sin(πψp) zasΓ(1− b + as) Γ(1− a[s] + as) pFp−1 ( 1− b + as 1− a[s] + as ; z ) = πGp,0p,p ( z b a ) − 1 π p∑ k=1 k 6=s sin(π(b− ak)) sin(π(a[k,s] − ak)) G2,p p,p ( z b ak, as,a[k,s] ) . (2.3) Proof. Formula (2.3) is a rewriting of [33, (5.40)]. Indeed, first compare [33, (2.44)] with the definition of G-function (2.1) to see how Nørlund’s ξn is expressed by Gp,0p,p. Alternatively, this connection follows on comparing the Mellin transforms (A.3) and [33, (2.18)]. Further, use [33, (5.45)] to expand Nørlund’s ϕs and [33, (5.7)] to express yk,s in terms of G2,p p,p (see also (2.10) below). � Remark 2.2. Formula (2.3) can now be viewed as the reflection of the fact that any p + 1 solutions must be linearly dependent so that any solution in the neighborhood of 0 can be expressed in terms of the fundamental set of solutions around 1. This formula extends the connection formula (1.2) for the Gauss hypergeometric function to which it reduces when p = 2. A closely related formula has been also discovered by Bühring in two papers [7] (for p = 3) and [8] (for general p). See formula (2.19) below. Remark 2.3. Marichev and Kalla reproduced formula [33, (5.40)] in [27, (11)] and gave expres- sions for its components in terms of G-function in [27, (27), (33), (36)]. Expansion of the solution Gp,0p,p in terms of the fundamental solutions around z = 0 that compliments (2.3) coincides with the well-known expansion obtained from the definition of G- function by the residue theorem. Namely, if the elements of the vector a are different modulo integers, formula (A.1) applied to the function Gp,0p,p takes the form (see also [26, formula (34)] and [36, 8.2.2.3]): Gp,0p,p ( z b a ) = p∑ k=1 zak Γ(a[k] − ak) Γ(b− ak) pFp−1 ( 1− b + ak 1− a[k] + ak z ) . (2.4) According to Theorem A.1(a) this formula holds for \|z\| < 1. 6 D. Karp and E. Prilepkina 2.2 Expansion of Gp,0 p,p in the neighborhood of unity Among many other results contained in [33] Nørlund showed that the series Gp,0p,p ( z b a ) = zak(1− z)ψp−1 Γ(ψp) ∞∑ n=0 gkp(n) (ψp)n (1− z)n (2.5) represents a solution in the neighborhood of z = 1 corresponding to the local exponent ψp − 1 (see (2.1) for the definition of ψp) if this number is not a negative integer. Formula (2.5) holds in the disk \|1− z\| < 1 for all −ψp /∈ N0 and each k = 1, 2, . . . , p. For −ψp = l ∈ N0 we have by taking limit in (2.5) (see [33, (1.34)]): Gp,0p,p ( z b a ) = zak ∞∑ n=0 gkp(n+ l + 1) n! (1− z)n, k = 1, 2, . . . , p. (2.6) The value of k in (2.5) and (2.6) can be chosen arbitrarily from the set {1, 2, . . . , p}. This choice affects the first factor zak and the coefficients gkp(n) while the left-hand side is of course independent of k. Let us present Nørlund’s formulas for the coefficients gkp(n). First, by applying the Frobenius ansatz to the differential equation (1.1) Nørlund demonstrated that gkp(n) satisfy the p-th order difference equation with polynomial coefficients given by p−1∑ i=0 Pp−i(n \|a,b)gkp(n+ i) = 0, n = 1, 2, . . . , (2.7) where Pm(z \|a,b) is a polynomial in z of degree m with coefficients dependent on the parame- ters a and b. It is expressed in terms of the polynomials Q(z) = p∏ i=1 (z − ai), R(z) = p∏ i=1 (z + 1− bi) as follows [33, (1.28)]: P1(z \|a,b) = p− 1 + z, Pj(z \|a,b) = (−1)j (p− j − 1)! ∆p−j−1Q(ψp + ak + z)− (−1)j (p− j)! ∆p−jR(ψp − 1 + ak + z) for j = 2, . . . , p− 1, and Pp(z \|a,b) = −(−1)pR(ψp − 1 + ak + z). Here the difference is understood as the forward difference ∆Q(z) = Q(z+1)−Q(z), ∆mQ(z) = ∆(∆m−1Q(z)). The initial values gkp(0), gkp(1), . . ., gkp(p− 1) for the recurrence (2.7) are found by solving the next triangular system: gkp(0) = 1, gkp(1) + P2(2− p \|a,b)gkp(0) = 0, 2gkp(2) + P2(3− p \|a,b)gkp(1) + P3(3− p \|a,b)gkp(0) = 0, 3gkp(3) + P2(4− p \|a,b)gkp(2) + P3(4− p \|a,b)gkp(1) + P4(4− p \|a,b)gkp(0) = 0, · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (p− 1)gkp(p− 1) + P2(0 \|a,b)gkp(p− 2) + P3(0 \|a,b)gkp(p− 3) +· · ·+ Pp(0 \|a,b)gkp(0) = 0. Hypergeometric Differential Equation and New Identities for the Coefficients 7 Another method to compute the coefficients gkp(n) discovered by Nørlund is the following re- currence in p [33, (2.7)]: g11(n) = { 1, n = 0, 0, n ≥ 1, gpp(n) = n∑ j=0 (bp − ak)n−j (n− j)! (ψp−1 + j)n−jg k p−1(n). (2.8) The last formula can be applied for any k = 1, 2, . . . , p− 1 without affecting the left-hand side. The value of gkp(n) for k 6= p can then be obtained by exchanging the roles of ap and ak in the resulting expression for gpp(n). Alternatively, Nørlund gives the following connection formula [33, (1.35)]: gkp(n) = n∑ j=0 (ak − al)n−j (n− j)! (ψp + j)n−jg l p(j), k, l = 1, . . . , p, k 6= l. Furthermore, he solved the above recurrence to obtain [33, (2.11)]: gpp(n) = ∑ 0≤j1≤j2≤···≤jp−2≤n p−1∏ m=1 (ψm + jm−1)jm−jm−1 (jm − jm−1)! (bm+1 − am)jm−jm−1 , (2.9) where, as before, ψm = ∑m i=1(bi − ai) and j0 = 0, jp−1 = n. This formula shows explicitly that gpp(n) does not depend on ap. Observe that • the coefficient gkp(n) is a symmetric polynomial in the components of b and a[k] (sepa- rately); • the summation in (2.9) is over all Young diagrams that fit n× (p− 2) box. Symmetry follows from expansion (2.5) and the invariance of G-function with respect to per- mutation the elements of a and b. In Section 3 below we find another way to compute the coefficients gkp(n) in terms of generalized Bernoulli polynomials and we further give explicit for- mulas for gkp(n), n = 1, 2, 3. Connection of these coefficients to combinatorics deserves further investigation. Remark 2.4. Formulas (2.3) and (2.5) have been reproduced by Marichev [26, (4), (17)] and Marichev and Kalla in [27, (12), (27)]. Surprisingly, these important formulas seem to have been largely overlooked in the special function literature and have never been included in any textbook on hypergeometric functions, except for the reference book [36] by Prudnikov, Brychkov and Marichev. However, even in this book the description of the behavior of Gp,0p,p(z) in the neighborhood of z = 1 contains an incorrect assertion in the case of non-positive integer ψp, see [36, Section 8.2.2.59]. None of the identities presented in Section 3 below are contained in [26, 27, 36]. Remark 2.5. Another very prolific line of research that involves G-function forms an important part of the statistics literature and began with 1932 paper of Wilks [44], where he observed that the moments of many likelihood ratio criteria in multivariate hypothesis testing are expressed in terms of product ratios of gamma functions. Wilks introduced two types of integral equations, of which “type B” is essentially equation (A.3) of the Appendix. He also noticed that the solution of “type B integral equation” represents the probability density of the product of independent beta distributed random variables. Wilks’ ideas were elaborated in dozens of papers that fol- lowed, mainly concerned with calculating and approximating the solution of “type B integral equation”. We will just mention a few key contributions, where an interested reader may find further references. In his 1939 paper [32] Nair derived the differential equation (1.1) satisfied 8 D. Karp and E. Prilepkina by the solution of “type B integral equation” thus demonstrating that Gp,0p,p satisfies (1.1). This happened long before Nørlund wrote his paper [33] and was also independent of Meijer’s work, although Meijer already introduced G-function as a linear combination of hypergeometric func- tions in 1936. Nair also was the first (among researchers in statistics) to apply the inverse Mellin transform to the right-hand side of (A.3) – the approach further developed by Consul in a series of papers between 1964 and 1969, where the connection to Meijer’s G-function was first observed. See [13] and references therein. Mellin transform technique was then utilized in a number of papers by Mathai who later rediscovered Nørlund’s coefficients in a form sim- ilar to (2.9) in [29]. Mathai’s and other contributions until 1973 are described in his survey paper [28]. In the same period Springer and Thompson independently expressed the densities of products and ratios of gamma and beta distributed random variables in terms of Meijer’s G-function, see [39]. Davis [14] presented the matrix form of the differential equation for Gp,0p,p and suggested the series solution similar to (2.5) with coefficients found by certain recursive procedure. Another noticeable contribution is due to Gupta and Tang who found two series ex- pansions for Gp,0p,p (again using “type B integral equation” terminology), one of them equivalent to (2.5), and rediscovered recurrence relation (2.8). See [40, 41] and references there. This line of research continues until today, as evidenced, for example, by a series of papers by Carlos Coelho with several co-authors, see [12] and references there. Furthermore, Charles Dunkl rediscovered Nørlund’s recurrence (2.7) in his preprint [16] dated 2013, where he again considers the prob- ability density of a product of beta distributed random variables. Independently, probability distribution with G-function density has been found to be the stationary distribution of certain Markov chains considered, for example, in actuarial science and is known as Dufresne law in this context. See [11, 15] and references therein. Let us also mention that G-function popped up recently in the random matrix theory as correlation kernel of a determinantal point process that governs singular values of products of M rectangular random matrices with independent complex Gaussian entries [1]. Curiously enough, none of these authors cited Nørlund’s work. Let us also mention that in our recent paper [23] conditions are given under which Gp,0p,p(e−x) is infinitely divisible distribution on [0,∞). 2.3 Expansion of G2,p p,p in the neighborhood of unity Formula [33, (5.7)] shows that Nørlund’s function y1,2(x) defined by [33, formula (5.2)] is ex- pressed in terms of Meijer’s G-function as follows y1,2(x) = G2,p p,p ( x 1− α1, . . . , 1− αp γ1, γ2, γ3, . . . , γp ) . (2.10) To define yk,s(x) the roles of γ1, γ2 are exchanged with those of γk, γs. Nørlund found several expansions of the functions yk,s(x) in hypergeometric polynomials which we cite below. Changing Nørlund’s notation to ours according to the rule x 7→ z, γi 7→ ai, αi 7→ 1− bi, formula [33, (5.3)] takes the form sinπ(as − ai)G2,p p,p ( z b as, ai,a[s,i] ) + sinπ(ai − ak)G2,p p,p ( z b ai, ak,a[i,k] ) + sinπ(ak − as)G2,p p,p ( z b ak, as,a[k,s] ) = 0 (2.11) for any distinct values of s, i, k ∈ {1, . . . , p}, where a[k,s] denotes the vector a with elements ak and as removed. Expansions [33, (5.20), (5.22), (5.23), (5.31)] written in terms of G-function take the form G2,p p,p ( z b ak, as,a[k,s] ) = zakΓ(1− b + ak)Γ(1− b1 + as)Γ(1− b2 + as) Γ(1− a[k,s] + ak)Γ(2 + ak + as − b1 − b2) Hypergeometric Differential Equation and New Identities for the Coefficients 9 × ∞∑ n=0 (1− b1 + ak)n(1− b2 + ak)n (2 + ak + as − b1 − b2)n p−1Fp−2 ( −n, 1− b[1,2] + ak 1− a[k,s] + ak z ) (2.12) = zakΓ(1− ak + as)Γ(1− b + ak) Γ(1− a[k,s] + ak) ∞∑ n=0 (1− b1 + ak)n (1− b1 + as)n+1 pFp−1 ( −n, 1− b[1] + ak 1, 1− a[k,s] + ak z ) (2.13) = zakΓ(1− b1 + as)Γ(1− b[1] + ak) Γ(1− a[k,s] + ak) ∞∑ n=0 (1− as + ak)n n!(1− b1 + ak + n) × pFp−1 ( −n, 1− b[1] + ak 1− a[k] + ak z ) (2.14) = zakπ(ak − as)Γ(1− b + ak) sin(π(ak − as))Γ(1− a[k] + ak) ∞∑ n=0 (1− as + ak)n (n+ 1)! p+1Fp ( −n, 1− b + ak 1, 1− a[k] + ak z ) . (2.15) The series in (2.12) converges in the disk \|z − 1\| < 1 if <(1 − b[1,2] + as) > 0, the series in (2.13), (2.14) converge in the same disk if <(1−b[1] +as) > 0 and, finally, (2.15) converges in \|z−1\| < 1 if <(1−b+as) > 0. In all cases, it is also required that none of the gamma functions in the numerator had poles. Further, Nørlund found two expansions of his function yk,s(x) in powers of 1− z, which we will need below. Written in our notation, formulas [33, (5.35), (5.36)] read G2,p p,p ( z b ak, as,a[k,s] ) = zas ∞∑ n=0 D[k,s] n (1− z)n, (2.16) where the coefficients D [k,s] n are given by D[k,s] n = Γ(1− b + ak)Γ(1− b1 + as + n)Γ(1− b2 + as + n) Γ(1− a[k,s] + ak)Γ(2 + ak + as − b1 − b2 + n)n! × ∞∑ j=0 (1− b1 + ak)j(1− b2 + ak)j j!(2 + ak + as − b1 − b2 + n)j p−1Fp−2 ( −j, 1− b[1,2] + ak 1− a[k,s] + ak ) (2.17) = Γ(1− b + ak)Γ(1− ak + as + n) Γ(1− a[k,s] + ak)n! ∞∑ j=0 (1− b1 + ak)j (1− b1 + as + n)j+1 × pFp−1 ( −j, 1− b[1] + ak 1, 1− a[k,s] + ak ) . (2.18) Here and below pFp−1 without an argument is understood as pFp−1(1). The series (2.16) con- verges in \|z − 1\| < 1 if <(1 − b[1,2] + as) > 0. The same condition suffices for convergence of (2.17), while (2.18) converges if <(1− b[1] + as) > 0. Uniqueness of power series coefficients implies equality of the coefficients in (2.17) and (2.18). For p = 2 this equality is nothing but Gauss formula for 2F1(1). For p = 3 we get after some simplifications and renaming variables 3F2 ( α1, α2, α3 β1, β2 ) = Γ(β2 − α1 − α2 + 1)Γ(β2) Γ(β2 − α1)Γ(β2 − α2) ∞∑ j=0 (α1)j (β2 − α2)j+1 3F2 ( −j, α2, β1 − α3 β1, 1 ) . 2.4 Connection to Bühring expansion of pFp−1 In his 1992 paper Bühring found the representation [8, Theorem 2] Γ(α) Γ(β) pFp−1 ( α β ; z ) = (1− z)ν ∞∑ n=0 fp(n\|α,β)(1− z)n + ∞∑ n=0 hp(n\|α,β)(1− z)n, (2.19) 10 D. Karp and E. Prilepkina where ν = p−1∑ k=1 βk − p∑ k=1 αk. He also derived explicit formulas for the coefficients fp(n\|α,β) and hp(n\|α,β) to be given below. On setting α = 1 − b + as, β = 1 − a[s] + as, ν = ψp − 1 and denoting fsp (n) ≡ fp(n\|1− b + as, 1− a[s] + as), hsp(n) ≡ hp(n\|1− b + as, 1− a[s] + as), Bühring’s formula takes the form Γ(1− b + as) Γ(1− a[s] + as) pFp−1 ( 1− b + as 1− a[s] + as ; z ) = (1− z)ψp−1 ∞∑ n=0 fsp (n)(1− z)n + ∞∑ n=0 hsp(n)(1− z)n. (2.20) Expansion of the form (2.19) is unique as long as ν is not an integer. Hence, in view of (2.4) and (2.17), identity (2.20) is equivalent to formula (2.3). From (2.3) we conclude that (1− z)ψp−1 ∞∑ n=0 fsp (n)(1− z)n = πz−as sin(πψp) Gp,0p,p ( z b a ) and (2.21) ∞∑ n=0 hsp(n)(1− z)n = −z−as π sin(πψp) p∑ k=1 k 6=s sin(π(b− ak)) sin(π(a[k,s] − ak)) G2,p p,p ( z b− as ak, as,a[k,s] ) (2.22) for each s ∈ {1, . . . , p}. Comparing (2.21) with (2.5) we arrive at fsp (n) = Γ(1− ψp) (ψp)n gsp(n), (2.23) where Euler’s reflection formula has been used. Thus, Bühring coefficients fp(n\|α,β) (denoted by gn(s) in [8]) are essentially the same as Nørlund coefficients gsp(n) (denoted by c (s) n,p in [33]). Explicit representation for the coefficients fp(n\|α,β) found in [8, (2.9), (2.16)] after some re- naming of variables and setting s = p can be put into the form fpp (n) = Γ(1− ψp) (ψp)n n∑ jp−2=0 (−1)jp−2(ψp + ap − bp−1 + jp−2)n−jp−2 (n− jp−2)! × jp−2∑ jp−3=0 · · · j3∑ j2=0 j2∑ j1=0 p−1∏ m=1 (ψm + jm−1)jm−jm−1 (jm − jm−1)! p−2∏ m=1 (bm − am+1)jm−jm−1 = Γ(1− ψp) (ψp)n ∞∑ j1,j2,...,jp−2=0 (ψp + ap − bp−1)n (ψp + ap − bp−1)jp−2 p−1∏ m=1 (ψm)jm(−jm)jm−1 (ψm)jm−1jm! × p−2∏ m=1 (bm − am+1)jm (1− bm + am+1 − jm)jm−1 , where ψm = m∑ i=1 (bi − ai) and j0 = 0, jp−1 = n. In view of (2.9), formula (2.23) then leads to the following (presumably new) transformation for multiple hypergeometric series n∑ jp−2=0 (ψp + ap − bp−1 + jp−2)n−jp−2 (−1)jp−2(n− jp−2)! Hypergeometric Differential Equation and New Identities for the Coefficients 11 × jp−2∑ jp−3=0 · · · j3∑ j2=0 j2∑ j1=0 p−1∏ m=1 (ψm + jm−1)jm−jm−1 (jm − jm−1)! p−2∏ m=1 (bm − am+1)jm−jm−1 = n∑ jp−2=0 · · · j3∑ j2=0 j2∑ j1=0 p−1∏ m=1 (ψm + jm−1)jm−jm−1 (jm − jm−1)! (bm+1 − am)jm−jm−1 , where j0 = 0 and jp−1 = n. For p = 3 this formula reduces to the identity 3F2 ( −n, α1, α2 β1, β2 ) = (β2 − α2)n (β2)n 3F2 ( −n, β1 − α1, α2 β1, 1− β2 + α2 − n ) , which is a guise of Sheppard’s transformation [2, Corollary 3.3.4], also rediscovered by Bühring [9, (4.1)]. For p = 4 the corresponding result is given by [9, (4.2)] n∑ k=0 (−k)n(α1)k(α2)k (β1)k(β2)kk! 3F2 ( −k, γ1, γ2 α1, α2 ) = (β2 − α2)n (β2)n n∑ k=0 (−k)n(α2)k(β1 − α1)k (β1)k(1 + α2 − β2 − n)kk! 3F2 ( −k, γ1, γ2 α2, 1 + α1 − β1 − k ) . In his paper [8] Bühring mentioned that the structure of (2.19) “was already given in the classic paper by Nørlund [33], but the coefficients were not all known”. In his subsequent joint work [10] with Srivastava is it said that the coefficients fsp (n) can be computed using Nørlund’s recurrence, “but it is desirable to get an explicit representation” which is indeed derived in [10] in terms of some limit relations. Explicit formulas for the coefficients hsp(n) have been first found in [8]. Adopted to our notation Bühring’s formulas [8, (2.9), (2.16)], [10, (2.13), (2.15)] read hpp(n) = −Γ(ψp)Γ(1− bp + ap + n)Γ(1− bp−1 + ap + n) (1− ψp)n+1Γ(ψp−1)Γ(ψp − bp−1 + ap)n! × ∞∑ jp−2=0 (ψp − n− 1)jp−2 (ψp−1)jp−2(ψp − bp−1 + ap)jp−2 × jp−2∑ jp−3=0 · · · j2∑ j1=0 p−2∏ m=1 (ψm + jm−1)jm−jm−1 (jm − jm−1)! (bm − am+1)jm−jm−1 , (2.24) where, as before, ψm = m∑ i=1 (bi − ai) and j0 = 0. Conditions for convergence for the outer series above have also been found in [8] and [10] and are given by <(1− bi + ap + n) > 0 for i = 1, 2, . . . , p− 2. Bühring and Srivastava found expressions for the coefficients fp(n\|α,β), hp(n\|α,β) as limits of hypergeometric polynomials [10, (3.13)]. They computed this limit for p = 3, 4, 5 yielding [10, (3.15), (3.17)] hs3(n) ≡ h3(n\|1− b + as, 1− a[s] + as) = − Γ(ψ3)Γ(1− b2 + as + n)Γ(1− b3 + as + n) (1− ψ3)n+1n!Γ(ψ3 − b2 + as)Γ(ψ3 − b3 + as) 3F2 ( ψ3 − 1− n, b1 − a[s] ψ3 − b[1] + as ) ,(2.25) hs4(n) = − Γ(ψ4)Γ(1− b3 + as + n)Γ(1− b4 + as + n) (1− ψ4)n+1n!Γ(ψ4 − b3 + as)Γ(ψ4 − b4 + as) 12 D. Karp and E. Prilepkina × ∞∑ k=0 (ψ4 − 1− n)k (b1 + b2 − ai1 − ai2)k(b1 + b2 − ai1 − ai3)k (ψ4 − b3 + as)k(ψ4 − b4 + as)k × 3F2 ( −k, b1 − ai1 , b2 − ai1 b1 + b2 − ai1 − ai2 , b1 + b2 − ai1 − ai3 ) , where {ai1 , ai2 , ai3} = a[s] (i.e., if s = 1 then {ai1 , ai2 , ai3} = {a2, a3, a4}, if s = 2 then {ai1 , ai2 , ai3} = {a1, a3, a4}, etc.). Expression for p = 5 is quite cumbersome and can be found in [10, (3.19)]. In spite of their non-symmetric appearance, both formulas are invariant with respect to permutation of the elements of b. Let us now cite the formulas for the coefficients gsp(n) for p = 2, 3, 4. For p = 2 the corre- sponding formulas can be read off formula (1.2). For both p = 2 and p = 3 expressions for gsp(n) have been found by Nørlund, see [33, (2.10)]. They are gs2(n) = (b1 − a3−s)n(b2 − a3−s)n n! , s = 1, 2, and gs3(n) = (ψ3 − b2 + as)n(ψ3 − b3 + as)n n! 3F2 ( −n, b1 − ai1 , b1 − ai2 ψ3 − b2 + as, ψ3 − b3 + as ) , s = 1, 2, 3, where {ai1 , ai2} = a[s]. Notwithstanding the non-symmetric appearance, the last formula is symmetric with respect to the elements of b. For p = 4 we convert the expression for f sp (n) calculated in [10, (3.17)] into expressions for gsp(n) using (2.23) and the necessary renaming of parameters. This yields gs4(n) = (ψ4 − b3 + as)n(ψ4 − b4 + as)n n! × n∑ k=0 (−n)k(b1 + b2 − ai1 − ai2)k(b1 + b2 − ai1 − ai3)k (ψ4 − b3 + as)k(ψ4 − b4 + as)k × 3F2 ( −k, b1 − ai1 , b2 − ai1 b1 + b2 − ai1 − ai2 , b1 + b2 − ai1 − ai3 ) , where {ai1 , ai2 , ai3} = a[s]. This formula is also invariant with respect to permutation of the elements of b. 3 Main results Having made these preparations, we are ready to formulate our main results. First, we give explicit formulas for the Nørlund’s coefficients gsp(1), gsp(2), gsp(3) with arbitrary p. Note that formula (2.9) contains p− 2 summations even for small n. Combined with (2.5) and gpp(0) = 1 our formulas essentially provide first four terms in asymptotic expansion of Gp,0p,p as z → 1. Then we derive a new way to compute the coefficients gsp(n) for all n in terms of generalized Bernoulli polynomials in Theorem 3.3. Theorems 3.5 and 3.10 contain new identities for the coefficients gsp(n), hsp(n) and D [k,s] n defined by expansions (2.5), (2.16) and (2.20), respectively. For small values of n these identities lead to new and known relations involving hypergeometric and trigonometric functions, in particular, to identity (1.3). Theorem 3.1. The first four coefficients of the expansion (2.5) are given by gpp(0) = 1, gpp(1) = p−1∑ m=1 (bm+1 − am)ψm, (3.1) Hypergeometric Differential Equation and New Identities for the Coefficients 13 gpp(2) = 1 2 p−1∑ m=1 (bm+1 − am)2(ψm)2 + p−1∑ k=2 (bk+1 − ak)(ψk + 1) k−1∑ m=1 (bm+1 − am)ψm (3.2) and gpp(3) = 1 6 p−1∑ m=1 (bm+1 − am)3(ψm)3 + 1 2 p−1∑ k=2 (bk+1 − ak)(ψk + 2) k−1∑ m=1 (bm+1 − am)2(ψm)2 + 1 2 p−1∑ k=2 (ψk + 1)2(bk+1 − ak)2 k−1∑ m=1 (bm+1 − am)ψm + p−1∑ n=3 (bn+1 − an)(ψn + 2) n−1∑ k=2 (bk+1 − ak)(ψk + 1) k−1∑ m=1 (bm+1 − am)ψm, (3.3) where, as before, ψm = m∑ j=1 (bj − aj). Proof. We note that for n = 1 summation in (2.9) is over the index sets of the form {j0, j1, . . ., jp−1} = {0, . . . , 0, 1, . . . , 1}, where the number of ones changes from 1 to p − 1. In view of this observation rearrangement of the formula (2.9) yields (3.1). Next, we prove (3.3). Summation in (2.9) is over all Young diagrams that fit 3 × (p − 2) box. We break all possible diagrams in four disjoint groups as follows (by definition j0 = 0, jp−1 = 3): (1) j0 = j1 = · · · = jm−1 = 0, jm = · · · = jp−1 = 3, m ∈ {1, 2, . . . , p− 1}, (2) j0 = j1 = · · · = jm−1 = 0, jm = · · · = jk−1 = 2, jk = · · · = jp−1 = 3, k ∈ {2, . . . , p− 1}, m ∈ {1, . . . , k − 1}, m < k, (3) j0 = j1 = · · · = jm−1 = 0, jm = · · · = jk−1 = 1, jk = · · · = jp−1 = 3, k ∈ {2, . . . , p− 1}, m ∈ {1, . . . , k − 1}, m < k, (4) j0 = · · · = jm−1 = 0, jm = · · · = jk−1 = 1, jk = · · · = jn−1 = 2, jn = · · · = jp−1 = 3, n ∈ {3, . . . , p− 1}, k ∈ {2, . . . , n− 1}, m ∈ {1, . . . , k − 1}, m < k < n. Summation over the first type of diagrams leads to the first term (3.3). Similarly, summation over the i-th type of diagrams leads to the i-th term (3.3) for i = 2, 3, 4. Analogous considerations lead to (3.2). � Remark 3.2. Exchanging the roles of ap and as formulas (3.1), (3.2) and (3.3) lead to ex- pressions for gsp(1), gsp(2), gsp(3). Since each coefficient gsp(n) is a symmetric polynomial in the elements of a[s] and b, such expressions can be expanded in terms of some basis of symmetric polynomials. In particular, denoting by ek(x1, . . . , xm) the k-th elementary symmetric polyno- mial of x1, . . . , xm, we get gsp(1) = e2(b)− e2(a[s]) + e1(a[s])(e1(a[s])− e1(b)) and (with the help of Mathematica) gsp(2) = 1 2 e1(a[s]) 4 − e1(a[s])3 + 1 2 e1(a[s]) 2 + 3 2 e1(a[s])e2(a[s]) + 1 2 e2(a[s]) 2 − 1 2 e2(a[s]) 14 D. Karp and E. Prilepkina − e1(a[s])2e2(a[s])− 1 2 e3(a[s]) + 1 2 e2(b)2 + 1 2 e2(b) + 1 2 e3(b)− 1 2 e1(a[s])e1(b) + 3 2 e1(a[s]) 2e1(b)− e1(a[s])3e1(b)− 1 2 e1(a[s])e1(b)2 + 1 2 e1(a[s]) 2e1(b)2 + e1(a[s]) 2e2(b) + 1 2 e1(b)e2(b)− e1(a[s])e1(b)e2(b)− e2(a[s])e2(b)− e1(b)e2(a[s]) + e1(a[s])e2(a[s])e1(b)− e1(a[s])e2(b). Next theorem gives a presumably new expression for Nørlund’s coefficients gkp(n) in terms of the generalized Bernoulli polynomials. Let us start by recalling that the Bernoulli–Nørlund (or the generalized Bernoulli) polynomial B(σ)k (x) is defined by the generating function [34, (1)]: tσext (et − 1)σ = ∞∑ k=0 B(σ)k (x) tk k! . In particular, B(1)k (x) = Bk(x) is the classical Bernoulli polynomial. Theorem 3.3. Coefficients gkp(n), defined in (2.5), are given by any of the following formulas gkp(n) = n∑ r=0 (−1)n−r(r + 1)n−r (n− r)! l̃rB(n+1) n−r (2− ak − ψp) = n∑ r=0 (−1)n−r(ψp + r)n−r (n− r)! lrB (n+ψp) n−r (1− ak). (3.4) Here l̃0 = 1 and l̃r, r ≥ 1, are found from the recurrence l̃r = 1 r r∑ m=1 q̃m l̃r−m, (3.5) where q̃m = (−1)m+1 m+ 1 Bm+1(ak + ψp − 1)− Bm+1(ak) + p∑ j=1 (Bm+1(aj)− Bm+1(bj))  . Similarly, the coefficients lr satisfy the recurrence relation lr = 1 r r∑ m=1 qmlr−m with l0 = 1 (3.6) and qm = (−1)m+1 m+ 1 p∑ j=1 ( Bm+1(aj)− Bm+1(bj) ) . Remark 3.4. It is known [22, Lemma 1] that the recurrences (3.5) and (3.6) can be solved to give the following explicit expressions for lr: lr = ∑ k1+2k2+···+rkr=r qk11 (q2/2)k2 · · · (qr/r)kr k1!k2! · · · kr! = r∑ n=1 1 n! ∑ k1+k2+···+kn=r n∏ i=1 qki ki . Hypergeometric Differential Equation and New Identities for the Coefficients 15 Similar formula is of course true for l̃r once we write q̃m instead of qm. Moreover, Nair [32, Section 8] found a determinantal expression for such solution which in our notation takes the form lr = det(Ωr) r! , Ωr = [ωi,j ] r i,j=1, ωi,j =  qi−j+1(i− 1)!/(j − 1)!, i ≥ j, −1, i = j − 1, 0, i < j − 1. Proof. The theorem is a corollary of an expansion of the H-function of Fox found in our recent paper [24]. Since Meijer’s G-function is a particular case of Fox’s H-function, formula (3.4) is a particular case of [24, Theorem 1] once we set p = q, A = B = (1, . . . , 1), ν = 1, µ = ψp, θ = ak − 1 in that theorem. � As before, we use the shorthand notation sin(π(b− ak)) = p∏ j=1 sin(π(bj − ak)), sin(π(a[k] − ak)) = p∏ j=1 j 6=k sin(π(aj − ak)), for the products and [a]j = a(a− 1) · · · (a− j + 1) for the falling factorial. Theorem 3.5. For each nonnegative integer m the following identities holds m∑ j=0 (−1)j j! p∑ k=1 [ak]jh k p(m− j) sin(π(b− ak)) sin(π(a[k] − ak)) = 0, (3.7) where the numbers hpp(n) are defined by (2.20) and given explicitly by (2.24), hkp(n) is obtained from hpp(n) by exchanging the roles of ap and ak; furthermore, m∑ j=0 (−1)j (ψp)m−jj! { [as]jg s p(m− j) sin(πψp)− p∑ k=1 [ak]jg k p(m− j) sin(π(b− ak)) sin(π(a[k] − ak)) } = 0, (3.8) where s ∈ {1, . . . , p} is chosen arbitrarily and the numbers gkp(n) are defined by expansion (2.5) and solve the recurrence (2.7) in n and the recurrence (2.8) in p. They are given explicitly by (2.9). Proof. Assume that the components of the vector a are distinct modulo 1. Substituting ex- pansion (2.20) into formula (2.4) and taking account of (2.23) we obtain Gp,0p,p ( z b a ) = (1− z)ψp−1 p∑ k=1 zak Γ(a[k] − ak)Γ(1− a[k] + ak) Γ(b− ak)Γ(1− b + ak) ∞∑ n=0 Γ(1− ψp) (ψp)n gkp(n)(1− z)n + p∑ k=1 zak Γ(a[k] − ak)Γ(1− a[k] + ak) Γ(b− ak)Γ(1− b + ak) ∞∑ n=0 hkp(n)(1− z)n = 1 π (1− z)ψp−1Γ(1− ψp) ∞∑ n=0 (1− z)n (ψp)n p∑ k=1 zak sin(π(b− ak)) sin(π(a[k] − ak)) gkp(n) + 1 π ∞∑ n=0 (1− z)n p∑ k=1 zak sin(π(b− ak)) sin(π(a[k] − ak)) hkp(n), 16 D. Karp and E. Prilepkina where we applied Euler’s reflection formula Γ(z)Γ(1 − z) = π/ sin(πz). Further, substitute Nørlund’s expansion (2.5) in place of G-function on the left-hand side and rearrange terms to get (1− z)ψp−1 ∞∑ n=0 (1− z)n { zasgsp(n) Γ(ψp + n) − Γ(1− ψp) (ψp)n p∑ k=1 zak sin(π(b− ak)) π sin(π(a[k] − ak)) gkp(n) } = ∞∑ n=0 (1− z)n p∑ k=1 zak sin(π(b− ak)) π sin(π(a[k] − ak)) hkp(n), where s ∈ {1, . . . , p} can be chosen arbitrarily. Denote for brevity γn,s(z) = zasgsp(n) Γ(ψp + n) − Γ(1− ψp) (ψp)n p∑ k=1 zak sin(π(b− ak)) π sin(π(a[k] − ak)) gkp(n), χn(z) = p∑ k=1 zak sin(π(b− ak)) π sin(π(a[k] − ak)) hkp(n), so that the above equality reduces to (1− z)ψp−1 ∞∑ n=0 (1− z)nγn,s(z) = ∞∑ n=0 (1− z)nχn(z). (3.9) Assume for a moment that ψp is not an integer. We know that both series in (3.9) converge in \|z − 1\| < 1. Furthermore, all functions γn,s(z) and χn(z) are analytic the same disk. This implies that (3.9) is only possible for all z in a disk centered at 1 if ∞∑ n=0 (1− z)nγn,s(z) ≡ 0 and ∞∑ n=0 (1− z)nχn(z) ≡ 0. (3.10) In terms of the functions χn(z) the claimed identity (3.7) takes the form m∑ j=0 (−1)j j! χ (j) m−j(1) = 0, (3.11) which we prove by induction in m. Letting z → 1 in the second identity in (3.10) we get χ0(1) = 0 which establishes our claim for m = 0. Next, suppose (3.11) holds for m = 0, 1, 2, . . . , r − 1. Divide the second identity in (3.10) by (1− z)r and expand each χn(z), n = 0, 1, . . . , r, in Taylor series around z = 1: 0 = r∑ n=0 (1− z)n−r { χn(1) + χ′n(1)(z − 1) + · · · + χ(r−n) n (z − 1)r−n/(r − n)! +O ( (z − 1)r−n+1 )} + ∞∑ n=r+1 (1− z)n−rχn(z) = r∑ k=0 (1− z)−k r−k∑ i=0 (−1)i i! χ (i) r−k−i(1) +O(z − 1) = r∑ i=0 (−1)i i! χ (i) r−k−i(1) +O(z − 1), where the last equality is by induction hypothesis. We now obtain (3.11) for m = r on letting z → 1 in this formula. It is immediate to check that the claimed identity (3.8) takes the form m∑ j=0 (−1)j j! γ (j) m−j,s(1) = 0, which can be demonstrated in a similar fashion starting with the first identity in (3.10). Hypergeometric Differential Equation and New Identities for the Coefficients 17 Finally, we remove the assumption that ψp is not an integer. The left-hand sides of (3.7) and (3.8) are analytic functions of, say, parameter b1 except for possible poles. Identities (3.7) and (3.8) for non-integer ψp then clearly imply by analytic continuation that these poles are removable and both identities hold for all ψp. � Corollary 3.6. Suppose for some i ∈ {1, 2, 3} inequality <(bi) < <(ak+1) holds for k ∈ {1, 2, 3}. Then 3∑ k=1 sin(π(b− ak))Γ(1− b[i] + ak) sin(π(a[k] − ak))Γ(ψ − b[i] + ak) 3F2 ( ψ − 1, bi − a[k] ψ − b[i] + ak ) = 0 (3.12) and 3∑ k=1 sin(π(b− ak))Γ(1− b[i] + ak) sin(π(a[k] − ak))Γ(ψ − b[i] + ak) { (1− b[i] + ak) 3F2 ( ψ − 2, bi − a[k] ψ − b[i] + ak ) − ak(2− ψ) 3F2 ( ψ − 1, bi − a[k] ψ − b[i] + ak )} = 0, (3.13) where ψ = 3∑ k=1 (bk − ak) and pFp−1 without argument is understood as pFp−1(1). Proof. Put p = 3. Identities (3.12) and (3.13) are now reformulations of (3.7) for m = 0 and m = 1, respectively. The coefficients hk3(0) and hk3(1) have been computed by formula (2.25). � The next corollary is a rewriting of (3.8) for m = 0 in view of gkp(0) = 1. Corollary 3.7. For any complex vectors a, b the following identity holds p∑ k=1 sin(π(b− ak)) sin(π(a[k] − ak)) = sin(πψp). (3.14) The right-hand side gives a continuous extension of the left-hand side if a[k]−ak contains integers. Remark 3.8. For p = 2 this is equivalent to Ptolemy’s theorem: if a quadrilateral is inscribed in a circle then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides, which can be written as sin(θ3 − θ1) sin(θ4 − θ2) = sin(θ2 − θ1) sin(θ4 − θ3) + sin(θ4 − θ1) sin(θ3 − θ2). Further details regarding the history behind the identity (3.14) can be found in the introduction and [21]. Corollary 3.9. For each m ∈ N0 and each p ∈ N the function Fp,m(a,b) = m∑ j=0 (−1)j j! [ak]j [ψp +m− 1]jg k p(m− j) = m∑ j=0 m−j∑ r=0 (−1)m−j(ψp + r)m−r j!(m− j − r)! lr[ak]jB (m−j+ψp) m−j−r (1− ak) (3.15) is independent of k and represents a symmetric polynomial in the components of a and b (separately). Here lr is defined by the recurrence relation (3.6). 18 D. Karp and E. Prilepkina Proof. Indeed, the first formula in (3.15) follows from (3.8) once we open the braces and apply the obvious relation 1/(ψp)m−j = [ψp +m− 1]j/(ψp)m. Substitution of (3.4) for gkp leads to the second formula. � The following theorem can be viewed as a new method for computing the coefficients hp(n\|α,β) in expansion (2.19) given by the multiple sum (2.24) by relating them to the numbers D [k,s] n given by the single sums (2.17) and (2.18). Theorem 3.10. For each nonnegative integer n and arbitrary s ∈ {1, . . . , p} the following identity holds hsp(n) = − 1 π sin(πψp) p∑ k=1 k 6=s sin(π(b− ak)) sin(π(a[k,s] − ak)) D[k,s] n , (3.16) where hsp(n) is defined by expansion (2.20) and given explicitly by (2.24), while D [k,s] n are given by (2.17) or (2.18). Moreover, for arbitrary distinct integers s, i, k from the set {1, 2, . . . , p} the following identity holds n∑ j=0 (−1)j j! ( [ai]j sin(π(as − ai))D[s,i] n−j + [ak]j sin(π(ai − ak))D [i,k] n−j + [as]j sin(π(ak − as))D [k,s] n−j ) = 0. (3.17) Proof. To prove (3.16) it suffices to substitute expansion (2.16) into formula (2.22) and equate coefficients. Identity (3.17) is a direct consequence of (2.11). � Corollary 3.11. For each n ∈ N0 the following identity holds true Γ(a3 − a1) Γ(b− a1)Γ(2 + a1 + a2 − b1 − b2 + n) 3F2 ( 1 + a1 − b1, 1 + a1 − b2, b3 − a3 1 + a1 − a3, 2 + a1 + a2 − b1 − b2 + n ) + Γ(a1 − a3) Γ(b− a3)Γ(2 + a2 + a3 − b1 − b2 + n) 3F2 ( 1 + a3 − b1, 1 + a3 − b2, b3 − a1 1 + a3 − a1, 2 + a2 + a3 − b1 − b2 + n ) = 1 Γ(2− ψ + n)Γ(ψ + a2 − b1)Γ(ψ + a2 − b2) 3F2 ( b3 − a1, b3 − a3, ψ − 1− n ψ + a2 − b1, ψ + a2 − b2 ) . Proof. For p = 3, s = 2 formula (3.16) takes the form h23(n) = − 1 π sin(πψ) ( sin(π(b− a1)) sin(π(a3 − a1) D[1,2] n + sin(π(b− a3)) sin(π(a1 − a3) D[3,2] n ) . (3.18) Now, write h23(n) according to (2.25) and exchange the roles of b1 and b3. Next, apply Chu– Vandermonde identity on the right-hand side of (2.17) to get D[k,s] n = Γ(1− b + ak)Γ(1− b1 + as + n)Γ(1− b2 + as + n) Γ(1− a[k,s] + ak)Γ(2 + ak + as − b1 − b2 + n)n! × 3F2 ( 1− b1 + ak, 1− b2 + ak, b3 − a[k,s] 2 + ak + as − b1 − b2 + n, 1− a[k,s] + ak ) . Substituting this into (3.18), applying Euler’s reflection formula for the gamma function and rearranging we get the claimed identity. � Hypergeometric Differential Equation and New Identities for the Coefficients 19 A Definition of Meijer’s G-function revisited Meijer’s G-function has been defined in the introduction, where we mentioned various aspects that need to be clarified in order that this definition be consistent. Most accurate information with proofs regarding G-function’s definition is contained, in our opinion, in the series of papers of Meijer himself [30], the paper [6] by Braaksma and in the first chapters of the books [35] and [25]. Further facts are scattered in the literature with most comprehensive collection being [36, Chapter 8], [3] and especially [31]. An accessible introduction to G-function can be found in a nice recent survey by Beals and Szmigielski [4]. In this paper we only deal with the function Gm,np,p . For convenience, we have gathered all the necessary information regarding its definition in the following theorem. Theorem A.1. Denote a∗ = m+ n− p, ψ = p∑ k=1 (bk − ak) and G(s) = Γ(b1 + s) · · ·Γ(bm + s)Γ(1− a1 − s) · · ·Γ(1− an − s) Γ(an+1 + s) · · ·Γ(ap + s)Γ(1− bm+1 − s) · · ·Γ(1− bp − s) . (a) If \|z\| < 1 then the integral in (2.1) converges for L = L− and Gm,np,p ( z a b ) = m∑ j=1 ∞∑ l=0 res s=bjl G(s)z−s, bjl = −bj − l. (A.1) If, in addition, a∗ > 0 and \| arg(z)\| < a∗π or a∗ = 0, <(ψ) < 0 and 0 < z < 1 then the integral in (2.1) also converges for L = Liγ and has the same value. (b) If \|z\| > 1 then the integral in (2.1) converges for L = L+ and Gm,np,p ( z a b ) = − n∑ i=1 ∞∑ k=0 res s=aik G(s)z−s, aik = 1− ai + k. If, in addition, a∗ > 0 and \| arg(z)\| < a∗π or a∗ = 0, <(ψ) < 0 and z > 1 then the integral in (2.1) also converges for L = Liγ and has the same value. Remark A.2. If \|z\| = 1 and <(ψ) < −1 then according to [25, Theorem 1.1] both integrals over L− and over L+ exist. If, in addition, a∗ > 0 and \| arg(z)\| < a∗π, then the integral over Liγ also exists. We found no proof in the literature that these integrals are equal. Proof. The claims regarding the contours L+ and L− have been demonstrated in [25, Theo- rems 1.1, 1.2]. They follow in a relatively straightforward manner from Stirling’s asymptotic formula for the gamma function and have been observed by Meijer himself in [30, Section 1]. If a∗ = 0, <ψ < 0 the result was proved by Kilbas and Saigo [25, Theorem 3.3]. For a∗ > 0 and \|z\| < 1 [30, Theorem D] states that the integrals over L− and Liγ coincide for any real γ which covers case (a) of the theorem. For the proof Meijer refers to [43, Section 14.51], where it is essentially demonstrated that lim R→∞ ∫ l−γ (R) m∏ j=1 Γ(s+ bj) n∏ i=1 Γ(1− s− ai) p∏ i=n+1 Γ(s+ ai) p∏ j=m+1 Γ(1− s− bj) z−sds = 0, (A.2) where l−γ (R) is the shortest arc of the circle \|s\| = R which connects the contours L− and Liγ (in fact, Whittaker and Watson proved a particular case, but the proof for the general case goes 20 D. Karp and E. Prilepkina along exactly the same lines). Since G(s) has no poles in the domain bounded by L− and Liγ this leads to equality of the integrals over L− and Liγ stated by Meijer. It remains to consider the case \|z\| > 1, L = Liγ . Denote by l+γ (R) the reflection of l−γ (R) with respect to Liγ . For \|z\| > 1 we get by changing s to −s ∫ l+γ (R) m∏ j=1 Γ(s+ bj) n∏ i=1 Γ(1− s− ai)z−s p∏ i=n+1 Γ(s+ ai) p∏ j=m+1 Γ(1− s− bj) ds = − ∫ l−γ (R) m∏ j=1 Γ(−s+ bj) n∏ i=1 Γ(1 + s− ai)zs p∏ i=n+1 Γ(−s+ ai) p∏ j=m+1 Γ(1 + s− bj) ds = − ∫ l−γ (R) n∏ i=1 Γ(s+ b′i) m∏ j=1 Γ(1− s− a′j) p∏ j=m+1 Γ(s+ a′j) p∏ i=n+1 Γ(1− s− b′i) (1/z)−sds. Here b′i = 1− ai, a′j = 1− bj . In view of (A.2), we immediately conclude that for \|z\| > 1 lim R→∞ ∫ l+γ (R) m∏ j=1 Γ(s+ bj) n∏ i=1 Γ(1− s− ai) p∏ i=n+1 Γ(s+ ai) p∏ j=m+1 Γ(1− s− bj) z−sds = 0, which implies that the integrals over L+ and Liγ coincide. � Remark A.3. If p > q (q > p) the integral in (2.1) exists for L = L+ (L = L−) and all complex z 6= 0 and is equal to the corresponding sum of residues [25, Theorems 1.1 and 1.2]. At the same time if a∗ > 0 and \| arg(z)\| < a∗π or a∗ = 0 and z > 0, z 6= 1, the integral in (2.1) also exists for L = Liγ . Most authors assume in this situation that Liγ can be deformed into L = L+ if p > q or L = L− if q > p without altering the value of the integral. However, we were unable to find any proof of this claim in the literature. Remark A.4. It follows from the above theorem that Gm,np,p (z) is analytic in the sector \| arg(z)\| < a∗π if a∗ > 0 (since the integral converges uniformly in z for L = Liγ), while for a∗ ≤ 0 we get two different analytic functions – one defined inside and the other outside of the unit circle, see [36, (8.2.2.7)]. In the proof of the above theorem we essentially used the next well-known reflection property of G-function Gm,np,q ( 1 z a b ) = Gn,mq,p ( z 1− b 1− a ) . It is important to note that by Theorem A.1(b) Gp,0p,p ( z b a ) = 0 for \|z\| > 1, Hypergeometric Differential Equation and New Identities for the Coefficients 21 which is, of course, different from the analytic continuation of the right-hand side of (2.4). The Mellin transform of Gp,0p,p exists if either <(ψ) > 0 or ψ = −m ∈ N0. In the former case [25, Theorem 2.2]∫ ∞ 0 xs−1Gp,0p,p ( x b a ) dx = ∫ 1 0 xs−1Gp,0p,p ( x b a ) dx = Γ(a + s) Γ(b + s) (A.3) for <(s) > −<(a). If ψ = −m ∈ N0 then [33, (2.28)]∫ ∞ 0 xs−1Gp,0p,p ( x b a ) dx = ∫ 1 0 xs−1Gp,0p,p ( x b a ) dx = Γ(a + s) Γ(b + s) − q(s) for <(s) > −<(a). Here q(s) is a polynomial of degree m given by q(s) = m∑ j=0 gkp(m− j)(s+ ak − j)j , k = 1, 2, . . . , p, where the coefficients gkp(n) are defined in expansion (2.5) and are given explicitly by (2.9). Note that gkp(n) depends on k while the polynomial q(s) is the same for each k ∈ {1, . . . , p}. Acknowledgements This work has been supported by the Russian Science Foundation under project 14-11-00022. We are also indebted to anonymous referees for a number of useful remarks that helped to improve the exposition. 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[44] Wilks S.S., Certain generalizations in the analysis of variance, Biometrika 24 (1932), 471–494. http://dx.doi.org/10.1016/0167-7152(84)90008-7 http://dx.doi.org/10.1016/0167-7152(84)90008-7 http://dx.doi.org/10.1111/j.1467-842X.1986.tb00588.x http://dx.doi.org/10.1111/j.1467-842X.1986.tb00588.x http://dx.doi.org/10.1007/BF01448236 http://dx.doi.org/10.1093/biomet/24.3-4.471 1 Introduction 2 Results of Nørlund and Bühring revisited 2.1 Fundamental solutions around unity 2.2 Expansion of Gp,0p,p in the neighborhood of unity 2.3 Expansion of G2,pp,p in the neighborhood of unity 2.4 Connection to Bühring expansion of pFp-1 3 Main results A Definition of Meijer's G-function revisited References

Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring

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