One-Step Recurrences for Stationary Random Fields on the Sphere
This paper develops operators for zonal functions on the sphere which preserve (strict) positive definiteness while moving up and down in the ladder of dimensions by steps of one. These fractional operators are constructed to act appropriately on the Gegenbauer polynomials {Cλn}.
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irk-123456789-1477442019-02-16T01:25:31Z One-Step Recurrences for Stationary Random Fields on the Sphere Beatson, R.K. W. zu Castell This paper develops operators for zonal functions on the sphere which preserve (strict) positive definiteness while moving up and down in the ladder of dimensions by steps of one. These fractional operators are constructed to act appropriately on the Gegenbauer polynomials {Cλn}. 2016 Article One-Step Recurrences for Stationary Random Fields on the Sphere / R.K. Beatson, W. zu Castell // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42A82; 33C45; 42C10; 62M30 DOI:10.3842/SIGMA.2016.043 http://dspace.nbuv.gov.ua/handle/123456789/147744 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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This paper develops operators for zonal functions on the sphere which preserve (strict) positive definiteness while moving up and down in the ladder of dimensions by steps of one. These fractional operators are constructed to act appropriately on the Gegenbauer polynomials {Cλn}. |
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Beatson, R.K. W. zu Castell |
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Beatson, R.K. W. zu Castell One-Step Recurrences for Stationary Random Fields on the Sphere Symmetry, Integrability and Geometry: Methods and Applications |
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Beatson, R.K. W. zu Castell |
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Beatson, R.K. |
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One-Step Recurrences for Stationary Random Fields on the Sphere |
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One-Step Recurrences for Stationary Random Fields on the Sphere |
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One-Step Recurrences for Stationary Random Fields on the Sphere |
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One-Step Recurrences for Stationary Random Fields on the Sphere |
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One-Step Recurrences for Stationary Random Fields on the Sphere |
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one-step recurrences for stationary random fields on the sphere |
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Інститут математики НАН України |
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2016 |
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One-Step Recurrences for Stationary Random Fields on the Sphere / R.K. Beatson, W. zu Castell // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 043, 19 pages
One-Step Recurrences for Stationary Random Fields
on the Sphere?
R.K. BEATSON † and W. ZU CASTELL ‡§
† School of Mathematics and Statistics, University of Canterbury,
Private Bag 4800, Christchurch, New Zealand
E-mail: r.beatson@math.canterbury.ac.nz
URL: http://www.math.canterbury.ac.nz/~r.beatson
‡ Scientific Computing Research Unit, Helmholtz Zentrum München,
Ingolstädter Landstraße 1, 85764 Neuherberg, Germany
E-mail: castell@helmholtz-muenchen.de
URL: http://www.helmholtz-muenchen.de/asc
§ Department of Mathematics, Technische Universität München, Germany
Received January 28, 2016, in final form April 15, 2016; Published online April 28, 2016
http://dx.doi.org/10.3842/SIGMA.2016.043
Abstract. Recurrences for positive definite functions in terms of the space dimension have
been used in several fields of applications. Such recurrences typically relate to properties of
the system of special functions characterizing the geometry of the underlying space. In the
case of the sphere Sd−1 ⊂ Rd the (strict) positive definiteness of the zonal function f(cos θ) is
determined by the signs of the coefficients in the expansion of f in terms of the Gegenbauer
polynomials {Cλn}, with λ = (d − 2)/2. Recent results show that classical differentiation
and integration applied to f have positive definiteness preserving properties in this context.
However, in these results the space dimension changes in steps of two. This paper develops
operators for zonal functions on the sphere which preserve (strict) positive definiteness while
moving up and down in the ladder of dimensions by steps of one. These fractional operators
are constructed to act appropriately on the Gegenbauer polynomials {Cλn}.
Key words: positive definite zonal functions; ultraspherical expansions; fractional integra-
tion; Gegenbauer polynomials
2010 Mathematics Subject Classification: 42A82; 33C45; 42C10; 62M30
1 Introduction
This paper develops operators for zonal functions on the sphere which preserve (strict) posi-
tive definitenesss while moving up and down in the ladder of dimensions by steps of one. The
operators provide tools for forming families of (strictly) positive definite zonal functions. Such
(strictly) positive definite zonal functions can be used as covariance models for estimating re-
gionalized variables and also for interpolation on spheres.
Within a deterministic context, zonal positive definite functions on the sphere have been used
for interpolation or approximation of scattered data (see [10, 11] and the references therein).
The standard ansatz in this setting is a linear combination of spherical translates of a fixed
(zonal) basis function. While the present paper could well have been stated within the context
of approximation on the sphere, we rather chose to provide a probabilistic framework, which is
to some extent is equivalent, i.e., the theory of regionalized variables.
?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:r.beatson@math.canterbury.ac.nz
http://www.math.canterbury.ac.nz/~r.beatson
mailto:castell@helmholtz-muenchen.de
http://www.helmholtz-muenchen.de/asc
http://dx.doi.org/10.3842/SIGMA.2016.043
http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 R.K. Beatson and W. zu Castell
Regionalized variables on spherical domains can nicely be modeled using random fields on
spheres [6, 15]. Such a random field is given through a set of random variables, Z(x) say, where
x ∈ Sd−1. Assuming the field to be Gaussian, i.e., for every n ∈ N, (Z(x1), . . . , Z(xn))T has
a multivariate Gaussian distribution for any choice of x1, . . . , xn ∈ Sd−1, the distribution can be
characterized by its first two moments.
Assuming second order (weak) stationarity, the covariance for an isotropic model is deter-
mined by a function
Cov
(
Z(x), Z(y)
)
= f(cos θ), x, y ∈ Sd−1,
where θ = θ(x, y) = arccos
(
xT y
)
is the geodesic distance between the points x and y on the
sphere Sd−1.
As a consequence of Kolmogorov’s extension theorem (see [5, Theorem 36.3]), the set of
isotropic Gaussian random fields can be identified with the set of zonal positive definite functions
on the sphere [12, 15]. We note in passing that Lévy named such processes Brownian motion.
Definition 1.1. A continuous function g : [0, π] → R is (zonal) positive definite on the sphe-
re Sd−1 if for all n ∈ N and all distinct point sets {x1, . . . , xn} on the sphere, the inequality
n∑
i,j=1
cicjg(θ(xi, xj)) ≥ 0
holds true for all c1, . . . , cn ∈ R. The function is (zonal) strictly positive definite on Sd−1 if the
inequality holds in the strict sense for all c1, . . . , cn ∈ R not vanishing simultaneously.
Although the natural distance on the unit sphere is an angle in [0, π], it is convenient for the
purpose of this paper to consider functions in x = cos θ ∈ [−1, 1], instead. Thus, by Λd−1 we
will denote the cone of all functions f ∈ C[−1, 1] such that f(cos · ) is positive definite on Sd−1.
Λ+
d−1 will denote the subcone of all strictly positive definite functions in Λd−1.
Gaussian random fields have been widely applied to statistically analyze spatial phenomena
[8, 16, 23]. In particular, kriging allows prediction of spatial variables from given samples at
arbitrary locations. The key ingredient for such an approach lies in determining a suitable
model for the covariance function of the spherical random field. Commonly, such a model can
be inferred from given data through fitting a parametric family of models (i.e., estimation of
the covariance).
Models for covariance functions have further been used for simulation of stationary random
fields. Matheron [17] suggested a method based on proper averaging of stationary random fields
on a lower dimensional space. In the Euclidean setting this so-called turning bands method works
as follows:
Given a stationary random field Z1 on the real line with covariance function C1 and a ran-
domly chosen direction ξ ∈ Sd−1, Zξ = Z1(x
T ξ) defines a stationary random field on Rd.
Averaging over all directions ξ leads to a stationary field on Rd the covariance function of
which, Cd say, relates to C1 via the so-called turning bands operator
Cd(t) = const
∫ ∞
0
(
1− τ2
) d−3
2
+
C1(tτ)dτ, t ∈ R+.
The turning bands operator represents one example out of a suite of operators, mapping radial
positive definite functions on Rd onto such functions on a higher or lower dimensional space.
Wendland [25], Wu [26] and Gneiting [13] used such operators to derive compactly supported
functions of a given smoothness. Recurrences for radial positive definite functions in general
have been studied by several authors [22, 27]. Due to Schoenberg’s characterization of radial
One-Step Recurrences for Stationary Random Fields on the Sphere 3
positive definite functions and the fact that scale mixtures of such functions preserve positive
definiteness, recurrence operators can be derived from corresponding relations between special
functions. In the case of radial functions on Rd, the appropriate fundamental relation is Sonine’s
first integral for Bessel functions of the first kind (see [27]).
In a recent paper [4] the authors applied similar operators to derive parametrized families of
suitable locally supported covariance models for stationary random fields on the sphere Sd−1.
These operators are based on properties of Gegenbauer polynomials, appearing in Schoenberg’s
characterization [24] of zonal positive definite functions on the sphere.
Theorem 1.2. Let λ = (d−2)/2, and consider a continuous function f on [−1, 1]. The function
f(cos · ) is positive definite on Sd−1, i.e., f ∈ Λd−1, if and only if f has an ultraspherical
expansion
f(x) ∼
∞∑
n=0
anC
λ
n(x), x ∈ [−1, 1], (1.1)
in which all the coefficients an are nonnegative, and the series converges at the point x = 1. If
this is the case, the series converges absolutely and uniformly on the whole interval.
Chen, Menegatto and Sun [7] showed that a necessary and sufficient condition for f ◦ cos to
be strictly positive definite on Sd−1, d ≥ 3, is that, in addition to the conditions of Theorem 1.2,
infinitely many of the Gegenbauer coefficients an with odd index, and infinitely many of those
with even index, are positive. In the case d = 2 the criteria is necessary but not sufficient
for f ◦ cos to be strictly positive definite. A characterization in this case has been given by
Menegatto, Oliveira & Peron [18], although the criterion is a little more involved (see also [3]
for further details on these issues).
In the same spirit as for the turning bands method, a zonal function defined on a lower
dimensional sphere Sd−κ can be lifted up to Sd−1 through averaging over the set of all copies
of Sd−κ contained in Sd−1. In [4] it is shown that the analogues for the sphere of Matheron’s
montée and descente operators (see [16]) for Rd are the operators
(I f)(x) =
∫ x
−1
f(u)du, x ∈ [−1, 1],
and
(D f)(x) = f ′(x), x ∈ [−1, 1].
Paralleling the behaviour of Matheron’s operators in the Euclidean case the operators move
in the ladder of dimensions by steps of two. Specifically, the I and D operators map zonal
positive definite functions f(cos ·) on Sd onto ones on Sd−2 and Sd+2, respectively (see [4] for
details). Therefore, the natural question arises, whether it would also be possible to proceed
through steps by one within the ladder of dimensions. While in the Euclidean case this could
be achieved using fractional differentiation and integration (see [27]), the situation is more
intriguing in the spherical setting. The reason lies in the fact that the characterizing special
functions for the sphere are polynomials, which are not preserved through fractional integration.
Thus, one has to work with combinations of fractional operators in order to guarantee that the
operators are mapping into the space of polynomials.
In the present paper, we provide a suite of four operators which can be used to define a clavier
(cf. [16]) for the sphere. The main results are given in Theorems 2.3, 2.4 and 2.8, below. We start
with introducing the appropriate fractional operators in the following section and studying their
action on ultraspherical expansions. The action of the operators on Gegenbauer polynomials
shown in the last section is derived using properties of hypergeometric 2F1-functions.
4 R.K. Beatson and W. zu Castell
2 Definition of the half-step operators
In the expansion (1.1) the dimension d appears in the parameter λ = (d−2)/2 of the Gegenbauer
polynomials. This relation between λ and d will be fixed throughout the paper.
From DCλn = 2λCλ+1
n−1 (cf. [9, 10.9(23)]) we see that classical differentiation and its inverse,
integration, alter the parameter λ by an integer. This is why the operators I and D traverse
the ladder of dimensions in steps of two (see [4]). At the same time, I and D change the degree
of polynomials by one. Therefore, in order to obtain a one-step operator in the dimension, we
have to consider fractional integration and differentiation, a fact which perfectly parallels the
Euclidean setting (see [20, 27]).
We are now ready to define the half-step operators and discuss their action on positive definite
functions on Sd−1.
Definition 2.1. For f ∈ L1[−1, 1] and λ ≥ 0, define
Iλ+f(x) = I
λ, 1
2
+ f(x) = (1 + x)−λ+
1
2
∫ x
−1
(x− τ)−
1
2 (1 + τ)λf(τ)dτ, (2.1)
Iλ−f(x) = I
λ, 1
2
− f(x) = (1− x)−λ+
1
2
∫ 1
x
(τ − x)−
1
2 (1− τ)λf(τ)dτ. (2.2)
Using these, we further define
Iλ+ = Iλ+ + Iλ− and Iλ− = Iλ+ − Iλ−. (2.3)
Apart from the additional factor (1±x)−λ+
1
2 in front of the integral and the weight (1± τ)λ,
the operators Iλ± are classical Riemann–Liouville fractional integrals of order 1
2 (cf. [21, De-
finition 2.1]) on the interval [−1, 1]. To define inverse operators, we use the corresponding
Riemann–Liouville fractional derivates (cf. [21, Definition 2.2]).
Definition 2.2. Let f be absolutely continuous on [−1, 1] and λ ≥ 0. Then
Dλ
+f(x) = D
λ, 1
2
+ f(x) = (1 + x)
d
dx
{
(1 + x)−λ
∫ x
−1
(x− τ)−
1
2 (1 + τ)λ−
1
2 f(τ)dτ
}
,
Dλ
−f(x) = D
λ, 1
2
− f(x) = (1− x)
d
dx
{
(1− x)−λ
∫ 1
x
(τ − x)−
1
2 (1− τ)λ−
1
2 f(τ)dτ
}
.
Using these, we further define
Dλ+ = Dλ
+ +Dλ
− and Dλ− = Dλ
+ −Dλ
−.
The main results of this paper are the following two theorems giving precise statements of the
dimension hopping and positive definiteness preserving properties of the operators Iλ± and Dλ±.
These are one-step analogues of Theorems 2.2 and 2.3 in [4]. Since in the light of Theorem 1.2
the statements can be considered as statements concerning ultraspherical expansions without
referring back to a sphere, we are considering m = d− 1 to be a positive integer.
Theorem 2.3. Let m be a positive integer and λ = (m− 1)/2.
(a) (i) Let f ∈ Λm+1, m ≥ 1. Then Iλ±f ∈ Λm.
(ii) Let f ∈ Λ+
m+1, m ≥ 2. Then Iλ±f ∈ Λ+
m.
(b) Let m ≥ 1, f ∈ Λ+
m+1 be nonnegative, and f have Gegenbauer expansion,
f ∼
∞∑
n=0
anC
λ+ 1
2
n ,
One-Step Recurrences for Stationary Random Fields on the Sphere 5
with all coefficients, {an}∞n=0, positive. Then Iλ+f is also nonnegative, Iλ+f ∈ Λ+
m, and all
the coefficients bn in the expansion
Iλ+f ∼
∞∑
n=0
bnC
λ
n ,
are positive.
Proof. The proofs for the statements are almost identical with those of the corresponding parts
of Proposition 2.2 in [4], provided that proper analogues for certain statements on Gegenbauer
polynomials are given. We therefore restrict ourselves to pointing out where adaptations of the
proof given in [4] are needed.
One of these details concerns the boundedness of the operators Iλ± as operators from C[−1, 1]
to C[−1, 1]. This follows from the definitions of Iλ± and Iλ± in equations (2.1), (2.2) and (2.3),
combined with the beta integrals∫ x
−1
(x− τ)−
1
2 (1 + τ)νdτ = (1 + x)ν+
1
2B
(
1
2 , ν + 1
)
= (1 + x)ν+
1
2
Γ(12)Γ(ν + 1)
Γ(ν + 3
2)
, (2.4)
and ∫ 1
x
(τ − x)−
1
2 (1− τ)νdτ = (1− x)ν+
1
2B
(
1
2 , ν + 1
)
= (1− x)ν+
1
2
Γ(12)Γ(ν + 1)
Γ(ν + 3
2)
.
Similarly, positivity of the operator Iλ+ follows from the definitions (2.1), (2.2) and (2.3). The
main ingredient thus remaining to be shown is the action of Iλ± on the Gegenbauer polyno-
mial C
λ+ 1
2
n . This part is given in Theorem 3.3, below. Note that in contrast to the operators
studied in [4], there is no need to deal with an extra constant in statements (i) and (ii). This
follows from Theorem 3.3, showing that the operators Iλ± do not introduce an additional con-
stant. �
Theorem 2.4. Suppose that f ∈ Λm, m ≥ 1, and let λ = (m − 1)/2. Then, if both functions
Dλ±f ∈ C[−1, 1], then Dλ±f ∈ Λm+1. If, in addition, f ∈ Λ+
m, then Dλ± ∈ Λ+
m+1.
Remark 2.5. Since the operators defined above can be seen as standard operators of fractional
integration/differentiation, classical results from fractional calculus can be applied. For example,
if (1 + τ)−1/2f(τ) ∈ Lipα for some α > 1
2 , in particular, if f ∈ Lipα and suppf ⊂ (−1, 1], then
by Theorem 19 in [14] D0
±f exists and is continuous. An analogous statement holds for general λ.
The proof of Theorem 2.4 depends heavily on a multiplier relationships between the Gegen-
bauer coefficients of f and those of Dλ±f . The details of these relationship, and the proof of
Theorem 2.4, will be deferred to the next subsection.
Let us finish the section with considering an example.
Example 2.6. Consider the operator Iλ+. In view of its definition (2.1) this operator maps
functions locally supported near one to functions locally supported near one. Also, since Iλ+ =
(Iλ+ + Iλ−)/2 this operator preserves (strict) positive definiteness by Theorem 2.3.
Note that by a change of variables∫ x
−1
(x− τ)−
1
2 (1 + τ)λf(τ)dτ = (x+ 1)λ+
1
2
∫ 1
0
(1− s)−
1
2 sλf
(
(x+ 1)s− 1
)
ds.
Therefore, if f were such that f
(
(x+ 1)s− 1
)
= (1− ys)−a, the integral becomes∫ 1
0
(1− s)−
1
2 sλ(1− ys)−ads =
Γ(λ+ 1)Γ
(
1
2
)
Γ
(
λ+ 3
2
) 2F1
[
a, λ+ 1
λ+ 3
2
∣∣∣∣ y], (2.5)
being a special case of Euler’s integral for hypergeometric functions (cf. [19, (15.6.1)]).
http://dlmf.nist.gov/15.6.E1
6 R.K. Beatson and W. zu Castell
Now consider the Cauchy family
ϕα,β(r) = (1 + rα)−
β
α , 0 < α ≤ 2, β > 0,
which is strictly positive definite on Rd for all d ≥ 1 (cf. [13]). Choosing α = 2 and restricting
the function ϕ2,β to the sphere, we obtain (setting τ = cos θ)
ϕβ(τ) = ϕ2,β
(√
2− 2 cos θ
)
= (3− 2τ)−
β
2 , β > 0.
Thus,
ϕβ
(
(x+ 1)s− 1
)
= 5−
β
2
(
1− 2
5(x+ 1)s
)−β
2 .
Therefore, from (2.5) we have that
Iλ+ϕβ(x) =
√
πΓ(λ+ 1)
5
β
2 Γ
(
λ+ 3
2
)(x+ 1)2F1
[
β
2 , λ+ 1
λ+ 3
2
∣∣∣∣ 25(x+ 1)
]
.
Since (see [19, (15.4.6)])
(1− z)−a = 2F1
[
a, b
b
∣∣∣∣ z], (2.6)
we can choose β = 2λ+ 3, yielding
Iλ+ϕ2λ+3(x) =
√
π
5
Γ(λ+ 1)
Γ
(
λ+ 3
2
)(x+ 1)(3− 2x)−(λ+1). (2.7)
Therefore, the function given in (2.7) is strictly positive definite on S2λ+1 by the remark at the
start of the example.
We can follow the same line of argument applying Iµ+ to Iλ+ϕ2λ+3. Again, the result is
a hypergeometric function. Interestingly, for µ = λ− 3
2 the series is of the form (2.6) resulting in
I
λ− 3
2
+
(
Iλ+ϕ2λ+3
)
(x) =
π
5
Γ
(
λ+ 1
2
)
Γ
(
λ+ 3
2
) (x+ 1)2(3− 2x)−(λ+ 1
2).
In general, for a function
gm,γ(x) = (x+ 1)m(3− 2x)−γ , m ∈ N0, γ > 0,
we obtain that
I
γ−m− 3
2
+ gm,γ(x) =
√
π
5
Γ
(
γ − 1
2
)
Γ(γ)
(x+ 1)m+1(3− 2x)−(γ− 1
2).
2.1 Ultraspherical expansions of f and Dλ
±f
The main results of this section will be Theorems 2.7 and 2.8 giving multiplier relationships
between the Gegenbauer coefficients of the (formal) series of f and those of the (formal) series
of Dλ+f and Dλ−f . These relationships will later be used to show that the operators Dλ+ and Dλ−
have the positive definiteness preserving properties given in Theorem 2.4.
The first statement shows that the operators D0
+, D0
− can be applied term by term to a Cheby-
shev series to obtain the formal Legendre series of D0
±f .
http://dlmf.nist.gov/15.4.E6
One-Step Recurrences for Stationary Random Fields on the Sphere 7
Theorem 2.7. Let f ∈ C[−1, 1] with (formal) Chebyshev series
f ∼
∞∑
n=0
anTn.
If both functions D0
±f ∈ C[−1, 1], then the (formal) Legendre series
D0
+f ∼
∞∑
n=0
bnPn
has coefficients
bn = (n+ 1)πan+1, n ∈ N0,
and the (formal) Legendre series
D0
−f ∼
∞∑
n=0
cnPn
has coefficients
cn = nπan, n ∈ N0.
Similar relations between coefficients in ultraspherical expansions hold for higher order
Gegenbauer polynomials.
Theorem 2.8. Let λ > 0, and let f ∈ C[−1, 1] have a (formal) Gegenbauer series
f ∼
∞∑
n=0
anC
λ
n .
If both functions Dλ±f ∈ C[−1, 1], then the (formal) Gegenbauer series
Dλ+f ∼
∞∑
n=0
bnC
λ+ 1
2
n
has coefficients
bn =
Γ(λ+ 1
2)
√
π
Γ(λ)
2(n+ 2λ+ 1)
n+ λ+ 1
an+1, n ∈ N0, (2.8)
and the (formal) Gegenbauer series
Dλ−f ∼
∞∑
n=0
cnC
λ+ 1
2
n ,
has coefficients
cn =
Γ(λ+ 1
2)
√
π
Γ(λ)
2n
n+ λ
an, n ∈ N0. (2.9)
8 R.K. Beatson and W. zu Castell
Remark 2.9. Theorem 2.7 is the limiting case of Theorem 2.8 under the limit
C0
n(x) = lim
λ→0+
1
λ
Cλn(x), n > 0, and C0
0 (x) = T0(x) = 1. (2.10)
Furthermore, we have the special cases
C0
n(x) =
2
n
Tn(x), n > 0,
while
C
1
2
n (x) = Pn(x) and C1
n(x) = Un(x), n ≥ 0.
Before proving the theorems we state the following technical lemma.
Lemma 2.10. For λ > 0, n ∈ N0 and x ∈ [−1, 1],
d
dx
{
(1 + x)Cλn(x)
}
= (n+ 1)Cλn(x) + 2
n−1∑
k=0
(k + λ)Cλk (x), (2.11)
d
dx
{
(1− x)Cλn(x)
}
= −(n+ 1)Cλn(x) + 2
n−1∑
k=0
(−1)k+n+1(k + λ)Cλk (x). (2.12)
Proof. Formula (2.12) can be obtained from equation (2.11) by using the reflection formula for
Gegenbauer polynomials
Cλn(−x) = (−1)nCλn(x). (2.13)
and the change of variables y = −x.
For the proof of formula (2.11) we will use two recurrences involving derivatives of Gegenbauer
polynomials which can be found, for example in [9, 10.9(35)]. For notational convenience we use
the (non-standard) notation Dλ
n(x) = d
dxC
λ
n(x) within the proof of the lemma. Then,
nCλn(x) = xDλ
n(x)−Dλ
n−1(x), and (2.14)(
1− x2
)
Dλ
n(x) =
(
1− x2
)
2λCλ+1
n−1(x) = (n+ 2λ− 1)Cλn−1(x)− nxCλn(x). (2.15)
Turn now to an induction proof of formula (2.11). The statement is clearly true for Cλ0 (x) = 1,
adopting the convention that the sum then is empty.
Now assume that n ∈ N and that the first statement is true for n− 1. Consider
d
dx
{
(1 + x)Cλn(x)
}
= Cλn(x) + (1 + x)Dλ
n(x). (2.16)
Using (2.15) and then (2.14) we obtain that
(1 + x)Dλ
n(x) =
(
1− x2
)
Dλ
n(x) + (1 + x)xDλ
n(x)
= (n+ 2λ− 1)Cλn−1(x)− nxCλn(x) + (1 + x)
(
nCλn(x) +Dλ
n−1(x)
)
= (n+ 2λ− 1)Cλn−1(x) + nCλn(x) + (1 + x)Dλ
n−1(x).
Therefore, applying (2.16),
d
dx
{
(1 + x)Cλn(x)
}
= (n+ 2λ− 1)Cλn−1(x) + (n+ 1)Cλn(x) + (1 + x)Dλ
n−1(x)
= (n+ 2λ− 1)Cλn−1(x) + (n+ 1)Cλn(x)
One-Step Recurrences for Stationary Random Fields on the Sphere 9
+
d
dx
{
(1 + x)Cλn−1(x)
}
− Cλn−1(x)
= (n+ 2λ− 2)Cλn−1(x) + (n+ 1)Cλn(x) +
d
dx
{
(1 + x)Cλn−1(x)
}
.
Using the induction hypothesis gives
d
dx
{
(1 + x)Cλn(x)
}
= (n+ 1)Cλn(x) + (n+ 2λ− 2)Cλn−1(x)
+ nCλn−1(x) + 2
n−2∑
k=0
(k + λ)Cλk (x)
= (n+ 1)Cλn(x) + 2
[
(n− 1) + λ
]
Cλn−1(x) + 2
n−2∑
k=0
(k + λ)Cλk (x),
which completes the proof. �
Proof of Theorem 2.7. Note that the continuity of the two functions D0
±f implies that of the
functions D0
±f . Then, proceeding by integration by parts, the coefficient bn of Pn in the formal
Legendre expansion of D0
+f is (2n+ 1)/2 times
H+ =
∫ 1
−1
Pn(x)
(
D0
+f
)
(x)dx
=
∫ 1
−1
Pn(x)(1 + x)
d
dx
(∫ x
−1
(x− t)−1/2(1 + t)−1/2f(t)dt
)
dx
=
[
Pn(x)(1 + x)
∫ x
−1
(x− t)−1/2(1 + t)−1/2f(t)dt
]1
−1
−
∫ 1
−1
∫ x
−1
(x− t)−1/2(1 + t)−1/2f(t)dt
d
dx
{
Pn(x)(1 + x)
}
dx.
In view of the formula [19, (5.12.1)],∫ x
−1
(x− t)−1/2(1 + t)−1/2dt = π,
for all −1 < x ≤ 1, the limit as x tends to −1 of the quantity within the square brackets vanishes.
Hence,
H+ = 2Pn(1)
∫ 1
−1
(
1− t2
)−1/2
f(t)dt
−
∫ 1
−1
∫ 1
t
(x− t)−1/2 d
dx
{Pn(x)(1 + x)}dx(1 + t)−1/2f(t)dt
= 2πa0 −
∫ 1
−1
∫ 1
t
(x− t)−1/2
[
(n+ 1)Pn(x) +
n−1∑
k=0
(2k + 1)Pk(x)
]
dx(1 + t)−1/2f(t)dt,
where the last step follows from an application of formula (2.11).
Noting the relationship (cf. [19, (18.17.46)])∫ 1
t
(x− t)−1/2Pk(x)dx =
1
(k + 1
2)
1√
1− t
[
Tk(t)− Tk+1(t)
]
, (2.17)
http://dlmf.nist.gov/5.12.E1
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10 R.K. Beatson and W. zu Castell
after some straightforward calculation, the double integral above turns into the form
H+ =
2n
2n+ 1
∫ 1
−1
(
1− t2
)− 1
2Tn(t)f(t)dt+
2(n+ 1)
2n+ 1
∫ 1
−1
(
1− t2
)− 1
2Tn+1(t)f(t)dt.
Therefore,
H+ =
n
2n+ 1
πan +
n+ 1
2n+ 1
πan+1.
Analogously, we define
H− =
∫ 1
−1
Pn(x)
(
D0
−f
)
(x)dx.
A similar integration by parts argument, but now using the formula (2.12), and the relationship
(cf. [19, (18.17.45)])∫ t
−1
(t− x)−1/2Pk(x)dx =
1
(k + 1
2)
1√
1 + t
[
Tk(t) + Tk+1(t)
]
, (2.18)
shows that
H− = − n
2n+ 1
πan +
n+ 1
2n+ 1
πan+1.
Since bn = 2n+1
2 (H+ +H−) we finally obtain
bn = (n+ 1)πan+1, n ∈ N0.
This completes the proof of the part of the theorem concerning D0
+f . The proof of the part of
the theorem concerning D0
−f is similar and will be omitted. �
The proof of Theorem 2.8 relies on a kind of fractional integration by parts. Before going
into details, we will state some technical lemmas.
Lemma 2.11. For λ ≥ 1/2, n ∈ N0 and x ∈ [−1, 1],
d
dx
{
(1 + x)λ+1(1− x)λC
λ+ 1
2
n (x)
}
= (1 + x)
(
1− x2
)λ−1
Qn+1(x), (2.19)
where
Qn+1(x) = (1− x)C
λ+ 1
2
n (x)− (n+ 1)
2λ+ n
2λ− 1
C
λ− 1
2
n+1 (x).
Proof. Note the formula (see [1, (22.13.2)] or [19, (18.17.1)] for the general Jacobi case)
n
(
1 +
n
2λ
)∫ x
0
(
1− t2
)λ− 1
2Cλn(t)dt = Cλ+1
n−1(0)−
(
1− x2
)λ+ 1
2Cλ+1
n−1(x),
which implies
d
dx
{(
1− x2
)λ
C
λ+ 1
2
n (x)
}
= −(n+ 1)
(
1 +
n+ 1
2(λ− 1
2)
)(
1− x2
)λ−1
C
λ− 1
2
n+1 (x), λ > 1/2.
http://dlmf.nist.gov/18.17.E45
http://dlmf.nist.gov/18.17.E1
One-Step Recurrences for Stationary Random Fields on the Sphere 11
Employing the relationship above, computing the derivative on the left hand side of (2.19)
yields
d
dx
{(
1− x2
)λ
C
λ+ 1
2
n (x)(1 + x)
}
= (1− x2)λCλ+
1
2
n (x) + (1 + x)
d
dx
{(
1− x2
)λ
C
λ+ 1
2
n (x)
}
=
(
1− x2
)λ
C
λ+ 1
2
n (x)− (1 + x)(n+ 1)
(
1 +
n+ 1
2λ− 1
)(
1− x2
)λ−1
C
λ− 1
2
n+1 (x)
= (1 + x)
(
1− x2
)λ−1{
(1− x)C
λ+ 1
2
n (x)− (n+ 1)
(
1 +
n+ 1
2λ− 1
)
C
λ− 1
2
n+1 (x)
}
, λ > 1/2.
Setting
Qn+1(x) = (1− x)C
λ+ 1
2
n (x)− (n+ 1)
(
1 +
n+ 1
2λ− 1
)
C
λ− 1
2
n+1 (x),
completes the proof when λ > 1/2. The limit relation (2.10) implies the result for λ = 1/2. �
Lemma 2.12. Let λ > 1, n ∈ N and x ∈ [−1, 1]. Then
(n+ 2λ− 1)Cλ−1n+1(x)− (n+ 2)Cλ−1n+2(x) = (2λ− 2)(1− x)
[
Cλn+1(x) + Cλn(x)
]
. (2.20)
Proof. Since d
dxC
λ
n(x) = 2λCλ+1
n−1(x), we have that
(2λ− 2)
[
Cλn+1(x) + Cλn(x)
]
=
d
dx
Cλ−1n+2(x) +
d
dx
Cλ−1n+1(x).
Using (cf. [9, 10.9(25), (15)])
d
dx
Cλn+1(x) = x
d
dx
Cλn(x) + (2λ+ n)Cλn(x)
and (
1− x2
) d
dx
Cλn(x) = (n+ 2λ)xCλn(x)− (n+ 1)Cλn+1(x)
we can proceed, obtaining
(1− x)(2λ− 2)
[
Cλn+1(x) + Cλn(x)
]
= (1− x)(2λ+ n− 1)Cλ−1n+1(x)
+ (n+ 2λ− 1)xCλ−1n+1(x)− (n+ 2)Cλ−1n+1(x),
from which the statement follows. �
The following proposition is the limit case of (2.20) taking the limit λ→ 1+ after multiplying
either side with 1/(λ− 1).
Proposition 2.13. Let n ∈ N0 and x ∈ [−1, 1]. Then
Tn+1(x)− Tn+2(x) = (1− x)
[
Un+1(x) + Un(x)
]
.
Proof. From [9, 10.11(3)] we have that
Tn+1(x) = Un+1(x)− xUn(x).
Furthermore, [9, 10.11(37)] yields
xUn+1(x) = Un+2(x)− Tn+2(x), 2Tn+2(x) = Un+2(x)− Un(x).
The claim follows from using these relations to rewrite the right hand side in terms of Chebyshev
polynomials of the first kind. �
12 R.K. Beatson and W. zu Castell
The following lemma states the analogues of equations (2.17) and (2.18) for general λ > 0.
Note that the first integral given in the lemma is a special case of Bateman’s integral (see [19,
(18.17.9)]). Using this, the statement follows from [19, (18.9.4)]. Rather than using Bateman’s
integral for Jacobi polynomials, we provide a proof staying within the family of Gegenbauer
polynomials, only.
Lemma 2.14. Let λ > 0, n ∈ N0 and x ∈ [−1, 1]. Then
1
Γ(λ)
∫ 1
x
(1− t)λ(t− x)−
1
2C
λ+ 1
2
n (t)dt
=
√
π
2Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)(1− x)λ−
1
2
[
(n+ 2λ)Cλn(x)− (n+ 1)Cλn+1(x)
]
,
and
1
Γ(λ)
∫ x
−1
(1 + t)λ(x− t)−
1
2C
λ+ 1
2
n (t)dt
=
√
π
2Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)(1 + x)λ−
1
2
[
(n+ 2λ)Cλn(x) + (n+ 1)Cλn+1(x)
]
.
Proof. From the definition we have that∫ 1
x
(1− t)λ(t− x)−
1
2C
λ+ 1
2
n (t)dt = (1− x)λ−
1
2 Iλ−C
λ+ 1
2
n (x),
and 2Iλ− = Iλ+ − Iλ−. We can therefore use Theorem 3.3 below to compute
2Iλ−C
λ+ 1
2
n (x) =
√
πΓ(λ)
Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)[(n+ 2λ)Cλn(x)− (n+ 1)Cλn+1(x)
]
,
from which the first statement follows. The second integral follows from the first by a change
of variables τ = −t and using the reflection formula (2.13). �
Remark 2.15. Taking the limit (2.10) readily leads to (2.17) and (2.18).
Corollary 2.16. Let λ ≥ 1, n ∈ N0 and x ∈ [−1, 1]. Then
1
Γ(λ)
∫ 1
x
(1− t)λ−1(t− x)−
1
2C
λ− 1
2
n+1 (t)dt
=
√
π
Γ
(
λ− 1
2
) (
n+ λ+ 1
2
)(1− x)λ−
1
2
[
Cλn+1(x) + Cλn(x)
]
,
and
1
Γ(λ)
∫ x
−1
(1 + t)λ−1(x− t)−
1
2C
λ− 1
2
n+1 (t)dt
=
√
π
Γ
(
λ− 1
2
) (
n+ λ+ 1
2
)(1 + x)λ−
1
2
[
Cλn+1(x)− Cλn(x)
]
.
Proof. The result follows from setting λ − 1 and n + 1 instead of λ and n, respectively in
Lemma 2.14 and applying Lemma 2.12. �
Remark 2.17. Formulæ (2.17) and (2.18) are the special cases for λ = 1.
http://dlmf.nist.gov/18.17.E9
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One-Step Recurrences for Stationary Random Fields on the Sphere 13
Proof of Theorem 2.8. The Fourier–Gegenbauer coefficients of the function Dλ+f are given
by
bn =
1
h
λ+ 1
2
n
∫ 1
−1
Dλ+f(t)C
λ+ 1
2
n (t)
(
1− t2
)λ
dt,
where
h
λ+ 1
2
n =
πΓ(2λ+ n+ 1)
22λn!
(
λ+ n+ 1
2
)
Γ2
(
λ+ 1
2
) .
Therefore,
h
λ+ 1
2
n bn =
∫ 1
−1
Dλ
+f(t)C
λ+ 1
2
n (t)
(
1− t2
)λ
dt+
∫ 1
−1
Dλ
−f(t)C
λ+ 1
2
n (t)
(
1− t2
)λ
dt.
Let us denote the first integral by H+ and the second by H−. From integration by parts it
follows that
H+ =
∫ 1
−1
(1 + x)
d
dx
{
(1 + x)−λ
∫ x
−1
(x− t)−
1
2 (1 + t)λ−
1
2 f(t)dt
}
C
λ+ 1
2
n (x)
(
1− x2
)λ
dx
=
[
(1 + x)−λ
∫ x
−1
(x− t)−
1
2 (1 + t)λ−
1
2 f(t)dt(1 + x)
(
1− x2
)λ
C
λ+ 1
2
n (x)
]1
−1
−
∫ 1
−1
(1 + x)−λ
∫ x
−1
(x− t)−
1
2 (1 + t)λ−
1
2 f(t)dt
d
dx
{
(1 + x)
(
1− x2
)λ
C
λ+ 1
2
n (x)
}
dx.
The constant term vanishes. Applying Lemma 2.11 we can decompose H+ into the sum of the
two integrals
I1 = −
∫ 1
−1
(1 + x)−λ
∫ x
−1
(x− t)−
1
2 (1 + t)λ−
1
2 f(t)dt
(
1− x2
)λ
C
λ+ 1
2
n (x)dx
= −
∫ 1
−1
∫ 1
t
(1− x)λ(x− t)−
1
2C
λ+ 1
2
n (x)dx(1 + t)λ−
1
2 f(t)dt
and
I2 = (n+ 1)
2λ+ n
2λ− 1
∫ 1
−1
(1 + x)−λ+1
∫ x
−1
(x− t)−
1
2 (1 + t)λ−
1
2 f(t)dt
(
1− x2
)λ−1
C
λ− 1
2
n+1 (x)dx
= (n+ 1)
2λ+ n
2λ− 1
∫ 1
−1
∫ 1
t
(1− x)λ−1(x− t)−
1
2C
λ− 1
2
n+1 (x)dx(1 + t)λ−
1
2 f(t)dt.
In an analogous way, we decompose H− into a sum of the integrals
I3 =
∫ 1
−1
∫ t
−1
(1 + x)λ(t− x)−
1
2C
λ+ 1
2
n (x)dx(1− t)λ−
1
2 f(t)dt
and
I4 = (n+ 1)
2λ+ n
2λ− 1
∫ 1
−1
∫ t
−1
(1 + x)λ−1(t− x)−
1
2C
λ− 1
2
n+1 (x)dx(1− t)λ−
1
2 f(t)dt.
The inner integrals in I1 and I3 can be computed using Lemma 2.14, whereas the corresponding
formulæ for I2 and I4 are stated in Corollary 2.16. Doing so, and using the definition of the
coefficients
an =
1
hλn
∫ 1
−1
f(t)Cλn(t)
(
1− t2
)λ− 1
2dt, where hλn =
πΓ(2λ+ n)
22λ−1n!(λ+ n)Γ2(λ)
,
14 R.K. Beatson and W. zu Castell
we obtain that
I1 =
π
3
2
22λn!Γ(λ)Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)
×
[
Γ(2λ+ n+ 1)
λ+ n+ 1
an+1 − (2λ+ n)
Γ(2λ+ n)
λ+ n
an
]
,
I2 =
π
3
2
22λn!Γ(λ)Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)
×
[
(2λ+ n)
Γ(2λ+ n+ 1)
λ+ n+ 1
an+1 + (2λ+ n)
Γ(2λ+ n)
λ+ n
an
]
,
I3 =
π
3
2
22λn!Γ(λ)Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)
×
[
(n+ 2λ)
Γ(2λ+ n)
λ+ n
an +
Γ(2λ+ n+ 1)
λ+ n+ 1
an+1
]
,
I4 =
π
3
2
22λn!Γ(λ)Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)
×
[
(2λ+ n)
Γ(2λ+ n+ 1)
λ+ n+ 1
an+1 − (2λ+ n)
Γ(2λ+ n)
λ+ n
an
]
.
Therefore,
H+ +H− =
π
3
2 Γ(2λ+ n+ 2)
22λ−1n!Γ(λ)Γ
(
λ+ 1
2
) (
n+ λ+ 1
2
)
(λ+ n+ 1)
an+1,
from which it follows that
bn =
2
√
πΓ
(
λ+ 1
2
)
Γ(λ)
2λ+ n+ 1
λ+ n+ 1
an+1.
This completes the proof of the part of the theorem concerning Dλ+f . Again, the proof of the
part concerning Dλ−f is similar and will therefore be omitted. �
Proof of Theorem 2.4. From the continuity assumption we have that the Gegenbauer series
of the functions Dλ±f are Abel summable. Since f is positive definite by hypothesis, Theorem 2.8
shows that the Gegenbauer coefficients of Dλ±f are non-negative. Hence the Gegenbauer series
restricted to x = 1 are series of non-negative terms. For such series of non-negative terms
Abel summability implies summability. Since C
λ+ 1
2
n attains its maximum at the point 1 we can
apply the Weierstraß M-test to show that the Gegenbauer series of Dλ±f converge uniformly
on [−1, 1]. Furthermore, the multipliers given in equations (2.8) and (2.9) preserve the sign of
the coefficients. Therefore, Dλ±f ∈ Λm+1 by Theorem 1.2. The statement about strict positive-
definiteness then follows from the same relation and the discussion on strict positive definiteness
in the paragraph following Theorem 1.2. �
3 The action of Iλ± and Dλ± on Gegenbauer polynomials
The claim that the operators Iλ± and Dλ± map Gegenbauer polynomials onto Gegenbauer polyno-
mials with a changed parameter is based upon contiguous relations for hypergeometric functions.
We will thus first state a proposition showing that the images of C
λ+ 1
2
n under the operators Iλ±
are hypergeometric polynomials.
One-Step Recurrences for Stationary Random Fields on the Sphere 15
Proposition 3.1. Let λ ≥ 0, n ∈ N0 and x ∈ [−1, 1]. Then
Iλ+C
λ+ 1
2
n (x) = cn,λ
λ+ 1
2
λ+ n+ 1
2
(1 + x) 2F1
[
−n, n+ 2λ+ 1
λ+ 1
2
∣∣∣∣ 1− x
2
]
, (3.1)
Iλ−C
λ+ 1
2
n (x) = cn,λ(1− x) 2F1
[
−n, n+ 2λ+ 1
λ+ 3
2
∣∣∣∣ 1− x
2
]
, (3.2)
where
cn,λ =
√
π(2λ+ 1)n
n!
Γ(λ+ 1)
Γ(λ+ 3
2)
.
Before proving the proposition let us state a technical lemma.
Lemma 3.2. Let λ > −1
2 , n ∈ N0 and x ∈ [−1, 1]. Then(
Iλ± 2F1
[
−n, n+ 2λ+ 1
λ+ 1
∣∣∣∣ 1±·
2
])
(x)
=
√
π
Γ(λ+ 1)
Γ(λ+ 3
2)
(1± x) 2F1
[
−n, n+ 2λ+ 1
λ+ 3
2
∣∣∣∣ 1± x
2
]
.
Proof. We will prove the statement in the ‘+’ case (the proof of ‘−’ case being analogous):(
Iλ+ 2F1
[
−n, n+ 2λ+ 1
λ+ 1
∣∣∣∣ 1 +·
2
])
(x)
= (1 + x)−λ+
1
2
∫ x
−1
(x− τ)−
1
2 (1 + τ)λ 2F1
[
−n, n+ 2λ+ 1
λ+ 1
∣∣∣∣ 1 + τ
2
]
dτ
=
n∑
k=0
(−n)k(n+ 2λ+ 1)k
(λ+ 1)kk!2k
(1 + x)−λ+
1
2
∫ x
−1
(x− τ)−
1
2 (1 + τ)λ+kdτ.
Using the beta integral (2.4) the expression above reduces to
n∑
k=0
(−n)k(n+ 2λ+ 1)k
(λ+ 1)kk!2k
Γ(12)Γ(λ+ k + 1)
Γ(λ+ k + 3
2)
(1 + x)k+1
=
√
π(1 + x)
n∑
k=0
(−n)k(n+ 2λ+ 1)kΓ(λ+ 1)
k!2k
(1 + x)k
(λ+ 3
2)kΓ(λ+ 3
2)
=
√
π
Γ(λ+ 1)
Γ(λ+ 3
2)
(1 + x) 2F1
[
−n, n+ 2λ+ 1
λ+ 3
2
∣∣∣∣ 1 + x
2
]
. �
The following proof of Proposition 3.1 uses an identity due to Pfaff (cf. [2, 2.3.14])
2F1
[
−m, b
c
∣∣∣∣ z] =
(c− b)m
(c)m
2F1
[
−m, b
b− c−m+ 1
∣∣∣∣ 1− z], m ∈ N0, (3.3)
which will occur frequently later on.
Proof of Proposition 3.1. From the representation of the Gegenbauer polynomials as hyper-
geometric functions [19, (18.5.9)], and the reflection formula (2.13) we have
Iλ+C
λ+ 1
2
n (x) = (−1)n
Γ(n+ 2λ+ 1)
n!Γ(2λ+ 1)
(
Iλ+ 2F1
[
−n, n+ 2λ+ 1
λ+ 1
∣∣∣∣ 1 +·
2
])
(x).
http://dlmf.nist.gov/18.5.E9
16 R.K. Beatson and W. zu Castell
Applying Lemma 3.2
Iλ+C
λ+ 1
2
n (x) = (−1)n
Γ(n+ 2λ+ 1)
n!Γ(2λ+ 1)
√
π
Γ(λ+ 1)
Γ(λ+ 3
2)
(1 + x) 2F1
[
−n, n+ 2λ+ 1
λ+ 3
2
∣∣∣∣ 1 + x
2
]
=
√
π(−1)n(2λ+ 1)n
n!
Γ(λ+ 1)
Γ(λ+ 3
2)
(1 + x) 2F1
[
−n, n+ 2λ+ 1
λ+ 3
2
∣∣∣∣ 1 + x
2
]
. (3.4)
Now (3.3) implies
2F1
[
−n, n+ 2λ+ 1
λ+ 3
2
∣∣∣∣ 1 + x
2
]
=
(−λ− n+ 1
2)n
(λ+ 3
2)n
2F1
[
−n, n+ 2λ+ 1
λ+ 1
2
∣∣∣∣ 1− x
2
]
. (3.5)
Substituting (3.5) into (3.4) noting that (a− n)n = (−1)n(1− a)n shows
Iλ+C
λ+ 1
2
n (x) =
√
π(2λ+ 1)n
n!
Γ(λ+ 1)
Γ(λ+ 3
2)
(λ+ 1
2)n
(λ+ 3
2)n
(1 + x) 2F1
[
−n, n+ 2λ+ 1
λ+ 1
2
∣∣∣∣ 1− x
2
]
=
√
π(2λ+ 1)n
n!
Γ(λ+ 1)
Γ(λ+ 3
2)
λ+ 1
2
λ+ n+ 1
2
(1 + x) 2F1
[
−n, n+ 2λ+ 1
λ+ 1
2
∣∣∣∣ 1− x
2
]
.
The proof of equation (3.2) is almost identical to that part of the proof of equation (3.1) up
to equation (3.4). It will therefore be omitted. �
Theorem 3.3. Let λ > 0 and n ∈ N0. Then
Iλ+C
λ+ 1
2
n (x) =
√
πΓ(λ)
Γ(λ+ 1
2)
n+ 2λ
n+ λ+ 1
2
Cλn(x), x ∈ [−1, 1], (3.6)
Iλ−C
λ+ 1
2
n (x) =
√
πΓ(λ)
Γ(λ+ 1
2)
n+ 1
n+ λ+ 1
2
Cλn+1(x), x ∈ [−1, 1]. (3.7)
Remark 3.4. Taking the limit of the last two relations as λ→ 0+ (see (2.10)) gives
I0+Pn(x) =
2
n+ 1
2
Tn(x), and I0−Pn(x) =
2
n+ 1
2
Tn+1(x), n ∈ N0.
Proof of Theorem 3.3. From (3.1) and (3.2) we obtain that
Iλ+C
λ+ 1
2
n (x) =
{
Iλ+ + Iλ−
}
C
λ+ 1
2
n (x)
=
2
√
π(2λ+ 1)n
(λ+ n+ 1
2)n!
Γ(λ+ 1)
Γ(λ+ 3
2)
{(
λ+
1
2
)(
1 + x
2
)
2F1
[
−n, n+ 2λ+ 1
λ+ 1
2
∣∣∣∣ 1− x
2
]
+
(
λ+ n+
1
2
)(
1− x
2
)
2F1
[
−n, n+ 2λ+ 1
λ+ 3
2
∣∣∣∣ 1− x
2
]}
.
An application of the contiguous relation [1, (15.2.25)] then shows
Iλ+C
λ+ 1
2
n (x) =
2
√
π(2λ+ 1)n
(λ+ n+ 1
2)n!
Γ(λ+ 1)
Γ(λ+ 3
2)
(
λ+
1
2
)
2F1
[
−n, n+ 2λ
λ+ 1
2
∣∣∣∣ 1− x
2
]
=
2
√
π(2λ+ 1)n
(λ+ n+ 1
2)n!
Γ(λ+ 1)
Γ(λ+ 1
2)
n!
(2λ)n
Cλn(x) =
√
π(2λ+ n)Γ(λ)
(λ+ n+ 1
2)Γ(λ+ 1
2)
Cλn(x),
which is equation (3.6).
The proof of equation (3.7) is analogous except that it uses the contiguous relation [19,
(15.5.16)]. Hence, it will be omitted. �
http://dlmf.nist.gov/15.5.E16
One-Step Recurrences for Stationary Random Fields on the Sphere 17
For the operators Dλ± we obtain the following result.
Theorem 3.5. Let λ > 0 and n ∈ N0. Then
Dλ+Cλn(x) =
√
πΓ(λ+ 1
2)
Γ(λ)
2(n+ 2λ)
n+ λ
C
λ+ 1
2
n−1 (x), x ∈ [−1, 1], (3.8)
Dλ−Cλn(x) =
√
πΓ(λ+ 1
2)
Γ(λ)
2n
n+ λ
C
λ+ 1
2
n (x), x ∈ [−1, 1]. (3.9)
Remark 3.6. Taking the limit of the last two relations as λ→ 0+ (see (2.10)) gives
D0
+Tn(x) = nπPn−1(x) and D0
−Tn(x) = nπPn(x), n ∈ N0.
Again, we first prove a preparatory proposition.
Proposition 3.7. Let λ > 0, n ∈ N0 and x ∈ [−1, 1]. Then
Dλ
+C
λ
n(x) = cn,λ
n(n+ 2λ)
n+ λ
1 + x
2
2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 1
∣∣∣∣ 1− x
2
]
, (3.10)
Dλ
−C
λ
n(x) = cn,λ
n(n+ 2λ)
λ+ 1
1− x
2
2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 2
∣∣∣∣ 1− x
2
]
, (3.11)
where
cn,λ =
√
π(2λ)n
n!
Γ(λ+ 1
2)
Γ(λ+ 1)
.
Proof. Consider the first equation
1
1 + x
Dλ
+C
λ
n(x) =
d
dx
{
(1 + x)−λ
∫ x
−1
(x− τ)−
1
2 (1 + τ)λ−
1
2Cλn(τ)dτ
}
=
(−1)n(2λ)n
n!
d
dx
{
(1 + x)−λ
∫ x
−1
(x− τ)−
1
2 (1 + τ)λ−
1
2 2F1
[
−n, n+ 2λ
λ+ 1
2
∣∣∣∣ 1 + τ
2
]
dτ
}
=
(−1)n(2λ)n
n!
d
dx
{
1
1 + x
I
λ− 1
2
+
(
2F1
[
−n, n+ 2λ
λ+ 1
2
∣∣∣∣ 1 +·
2
])
(x)
}
.
Applying Lemma 3.2
1
1 + x
Dλ
+C
λ
n(x) =
(−1)n(2λ)n
n!
√
π
Γ(λ+ 1
2)
Γ(λ+ 1)
d
dx
2F1
[
−n, n+ 2λ
λ+ 1
∣∣∣∣ 1 + x
2
]
.
The formula (cf. [19, (15.2.1)])
d
dx
2F1
[
a, b
c
∣∣∣∣x] =
(a)1(b)1
(c)1
2F1
[
a+ 1, b+ 1
c+ 1
∣∣∣∣x]
then implies
Dλ
+C
λ
n(x) =
(−1)n(2λ)n
n!
√
π
Γ(λ+ 1
2)
Γ(λ+ 1)
1 + x
2
(−n)(n+ 2λ)
(λ+ 1)
× 2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 2
∣∣∣∣ 1 + x
2
]
=
√
π(2λ)nΓ(λ+ 1
2)
n!Γ(λ+ 1)
(−1)n+1 1 + x
2
(n)(n+ 2λ)
(λ+ 1)
http://dlmf.nist.gov/15.5.E1
18 R.K. Beatson and W. zu Castell
× 2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 2
∣∣∣∣ 1 + x
2
]
. (3.12)
Now from equation (3.3)
2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 2
∣∣∣∣ 1 + x
2
]
=
(−1)n−1(1 + λ)
n+ λ
2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 1
∣∣∣∣ 1− x
2
]
.
Substituting into equation (3.12) gives equation (3.10).
The proof of equation (3.11) is almost identical to that of equation (3.10). It will therefore
be omitted. �
Proof of Theorem 3.5. From (3.10) and (3.11)
Dλ+Cλn(x) =
√
π(2λ)n
n!
Γ(λ+ 1
2)
Γ(λ+ 1)
n(n+ 2λ)
(n+ λ)(λ+ 1)
×
{
(λ+ 1)
(
1 + x
2
)
2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 1
∣∣∣∣ 1− x
2
]
+ (n+ λ)
(
1− x
2
)
2F1
[
−n+ 1, n+ 2λ+ 1
λ+ 2
∣∣∣∣ 1− x
2
]}
.
An application of the contiguous relation [1, (15.2.25)] then shows
Dλ+Cλn(x) =
√
π(2λ)n
n!
Γ(λ+ 1
2)
Γ(λ+ 1)
n(n+ 2λ)
(n+ λ)
2F1
[
−n+ 1, n+ 2λ
λ+ 1
∣∣∣∣ 1− x
2
]
,
and rewriting the hypergeometric function on the right as a Gegenbauer polynomial gives (3.8).
The proof of equation (3.9) is analogous except that it uses contiguous relation [19, (15.5.16)].
Hence, it will be omitted. �
Acknowledgements
The authors thank the editor and the referees for their helpful suggestions. WzC acknowledges
support from a University of Canterbury Visiting Erskine Fellowship. RKB is grateful for the
hospitality provided by the Helmholtz Zentrum München.
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http://dx.doi.org/10.1006/jmva.2001.2056
http://dx.doi.org/10.1007/BF01171116
http://dx.doi.org/10.1007/BF02843693
http://dx.doi.org/10.1016/j.camwa.2006.04.006
http://dlmf.nist.gov/
http://dx.doi.org/10.1007/s00477-006-0079-9
http://dx.doi.org/10.1016/0377-0427(96)00047-7
http://dx.doi.org/10.1007/978-3-642-17086-7_2
http://dx.doi.org/10.1215/S0012-7094-42-00908-6
http://dx.doi.org/10.1007/BF02123482
http://dx.doi.org/10.1007/BF03177517
http://dx.doi.org/10.1016/S0022-247X(02)00101-4
1 Introduction
2 Definition of the half-step operators
2.1 Ultraspherical expansions of f and D f
3 The action of I and D on Gegenbauer polynomials
References
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