Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials
By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established.
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irk-123456789-1657332020-02-17T01:26:16Z Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials Atakishiyev, N.M. Klimyk, A.U. Статті By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established. 3a допомогою двох операторів, зображуваних матрицями Якобі, введено сім'ю q-ортогональних многочленів, що є дуальними по відношенню до альтернативних q-многочленів Шарльє. Для цих многочленів отримано дискретне співвідношення ортогональності та властивість повноти. 2005 Article Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials / N.M.Atakishiyev, A.U. Klimyk // Український математичний журнал. — 2005. — Т. 57, № 5. — С. 614–621. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165733 517.986 + 517.58 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Atakishiyev, N.M. Klimyk, A.U. Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials Український математичний журнал |
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By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established. |
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Article |
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Atakishiyev, N.M. Klimyk, A.U. |
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Atakishiyev, N.M. Klimyk, A.U. |
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Atakishiyev, N.M. |
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Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials |
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Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials |
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Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials |
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Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials |
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Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials |
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jacobi matrix pair and dual alternative q-charlier polynomials |
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Інститут математики НАН України |
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2005 |
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Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials / N.M.Atakishiyev, A.U. Klimyk // Український математичний журнал. — 2005. — Т. 57, № 5. — С. 614–621. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
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AT atakishiyevnm jacobimatrixpairanddualalternativeqcharlierpolynomials AT klimykau jacobimatrixpairanddualalternativeqcharlierpolynomials |
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2025-07-14T19:44:47Z |
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1837652811931713536 |
| fulltext |
UDC 517.986+517.58
N. M. Atakishiyev (Inst. Math., UNAM, Mexico),
A. U. Klimyk (Inst. Theor. Phys. Nat. Acad. Sci. Ukraine, Kyiv)
JACOBI MATRIX PAIR AND DUAL ALTERNATIVE
q-CHARLIER POLYNOMIALS
*
PARA MATRYC| QKOBI TA DUAL|NI AL|TERNATYVNI
q-MNOHOÇLENY ÍARL|{
By using two operators, representable by Jacobi matrices, we introduce a family of q-orthogonal
polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete
orthogonality relation and completeness property for these polynomials are obtained.
Za dopomohog dvox operatoriv, zobraΩuvanyx matrycqmy Qkobi, vvedeno sim’g q-ortohonal\nyx
mnohoçleniv, wo [ dual\nymy po vidnoßenng do al\ternatyvnyx q-mnohoçleniv Íarl\[. Dlq
cyx mnohoçleniv otrymano dyskretne spivvidnoßennq ortohonal\nosti ta vlastyvist\ povnoty.
1. Introduction. It is well known that the characteristic properties of orthogonal
polynomials are interwoven with spectral properties of symmetric operators, which can
be represented in some basis by a Jacobi matrix, and with the classical moment
problem (see, for example, [1 – 3]). Namely, the spectrum of an operator, represented
by a Jacobi matrix, is determined by an orthogonality measure for corresponding ortho-
gonal polynomials. If orthogonal polynomials admit many orthogonality relations,
then the corresponding symmetric operator is not self-adjoint and it has infinitely many
self-adjoint extensions. These extensions are determined by orthogonality measures for
the appropriate orthogonal polynomials.
Contrary to orthogonal polynomials of the hypergeometric type (Wilson, Jacobi,
Laguerre and so on), basic hypergeometric polynomials (or q-orthogonal polynomials)
are not so deeply understood yet. These polynomials are collected in the q-analogue of
the Askey-scheme of orthogonal polynomials (see, for example, [4]), which starts with
Askey – Wilson polynomials and q-Racah polynomials, introduced in [5] and [6],
respectively. The importance of these polynomials is magnified by the fact that they
are closely related to the theory of quantum groups. As an instance of such connection
we refer to a paper [7], in which Al-Salam – Chihara q-orthogonal polynomials have
been employed to construct locally compact quantum group SUq (1, 1). Another appli-
cation of q-orthogonal polynomials is related to the theory of q-difference equations,
which often surface in contemporary theoretical and mathematical physics.
The purpose of the present paper is to study the duality properties of alternative q-
Charlier polynomials. We shall show below that this leads to novel type of q-orthogo-
nal polynomials (expressed in terms of the basic hypergeometric function 3φ0 ) and a
discrete orthogonality relation for them. To achieve this, we essentially use two
operators I1 and J, which are certain representation operators for the quantum
algebra Uq ( su1,1 ) with a lowest weight (however, we do not employ explicitly the
theory of representations in what follows). An orthogonality relation for a set of
orthogonal polynomials and the spectral measure for the associated symmetric (self-
adjoint) operator, representable by a Jacobi matrix, are closely interrelated. But the
problem usually arises how to find this orthogonality relation (or spectral measure).
We propose to use for this purpose a second operator, also representable (with respect
to another basis) by a Jacobi matrix. By employing these two operators, one is able to
find orthogonality relations for both alternative q-Charlier polynomials and their duals.
The first operator I1 (which is a Hilbert – Schmidt operator and, therefore, has the
* This work was performed during a visit of the second author (AUK) to the Institute of Mathematics,
UNAM, Mexico. The participation of the first author (NMA) has been supported in part by the
DGAPA-UNAM project IN102603-3 “Óptica Matemática”.
© N. M. ATAKISHIYEV, A. U. KLIMYK, 2005
614 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
JACOBI MATRIX PAIR AND DUAL ALTERNATIVE … 615
discrete spectrum) is related to the three-term recurrence relation for alternative q-
Charlier polynomials. We diagonalize the operator I1 and obtain two bases in the
Hilbert space: an initial basis and a basis of eigenvectors of I1. The initial basis is
orthonormalized. It is a problem to normalize the second basis. We normalize it by
means of the second operator J. These orthonormalized bases are connected by an
orthogonal matrix A (since its matrix elements are real). Now the orthogonality
relations for rows and columns of this matrix lead to the orthogonality relations for
alternative q-Charlier polynomials and for the functions, which are dual to these
polynomials (note that the orthogonality relation for alternative q-Charlier polynomials
is given in [4], Section 3.22, but no proof of it was published; as is written in [4], no
other references to these polynomials are known). We extract from the latter functions
a dual set of polynomials and obtain a discrete orthogonality relation for them. The
unitarity of the matrix A, A A
–
1 = A
–
1
A = E, proves the completeness property of
dual alternative q-Charlier polynomials in the corresponding Hilbert space L
2.
Throughout the sequel we always assume that q is a fixed positive number such
that q < 1. We use (without additional explanation) notations of the theory of special
functions and the standard q-analysis (see, for example, [8] or [9]).
2. Pair of operators (((( I1, J )))) . Let H be a separable complex Hilbert space with
an orthonormal basis n〉 , n = 0, 1, 2, … . We define on H two operators. The first
one, denoted as qJ0 and taken from the theory of representations of quantum groups,
acts upon the basis elements as q n q nJ n0 〉 〉= . The second operator, denoted as I1 ,
is given by the formula
I n1 〉 = a n a n b nn n n+ + − +〉 〉 〉−1 11 , (1)
an = – ( )
( )( )
( ) ( )( )
/aq
q aq
aq aq aq
n
n n
n n n
3 1 1 2
1
2 1 2 2 2
1 1
1 1 1
+
+
+ +
− +
+ + +
,
bn = q
aq
aq aq
aq
q
aq aq
n
n
n n
n
n
n n
1
1 1
1
1 12 2 1
1
2 1 2
+
+ +
+ −
+ +
+
−
−( )( ) ( )( )
,
where a is any fixed positive number. Clearly, I1 is a symmetric operator.
Since an → 0 and bn → 0 when n → ∞ , the operator I1 is bounded.
Therefore, we assume that it is defined on the whole Hilbert space H . For this reason,
I1 is a self-adjoint operator. Let us show that I1 is a Hilbert – Schmidt operator. For
the coefficients an and bn from (1), we have a a qn n+ →1
3 2/ / and b b qn n+ →1 /
when n → ∞ . Since 0 < q < 1, for the sum of all matrix elements of the operator I1
in the basis n〉 , n = 0, 1, 2, … , we have ( )2a bn nn
+ < ∞∑ and, therefore,
( )2 2 2a bn nn
+ < ∞∑ . Thus, I1 is a Hilbert – Schmidt operator. This means that the
spectrum of I1 is discrete and has a single accumulation point at 0. Moreover, the
spectrum of I1 is simple, since I1 is representable by a Jacobi matrix with an ≠ 0 (see
[2], Chapter VII).
To find eigenfunctions ξλ of the operator I1 , I1ξ λξλ λ= , we set ξ λ =
= β λnn
n( ) 〉∑ , where β λn( ) are appropriate numerical coefficients. Acting by the
operator I1 upon both sides of this relation, one derives that
β λn n n n
n
a n a n b n( ) + + − +( )〉 〉 〉−
=
∞
∑ 1 11
0
= λ β λn
n
n( ) 〉
=
∞
∑
0
,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
616 N. M. ATAKISHIYEV, A. U. KLIMYK
where an and bn are the same as in (1). Collecting in this identity all factors, which
multiply n〉 with fixed n, one derives the recurrence relation for the coefficients
βn ( λ ) :
β λ β λ β λn n n n n na a b+ − −+ +1 1 1( ) ( ) ( ) = λβ λn( ).
The substitution
βn ( λ ) =
( ; ) ( )
( ; ) ( )( )
( )
/
/
( )/− +
+
′− +a q aq
q q a a q
qn
n
n
n
n n
n
1
1
2 1 2
3 4 β λ
reduces this relation to the following one
– A C A Cn n n n n n n
′ − ′ + + ′
+ −β λ β λ β λ1 1( ) ( ) ( ) ( ) = λβ λ′
n( ),
An = q
aq
aq aq
n
n
n n
1
1 12 2 1
+
+ + +( )( )
, Cn = aq
q
aq aq
n
n
n n
2 1
2 1 2
1
1 1
−
−
−
+ +( )( )
.
This is the recurrence relation for the alternative q-Charlier polynomials
K a qn( ; ; )λ : = 2 1 0φ λ( , ; ; , )q aq q qn n− −
(see formulas (3.22.1) and (3.22.2) in [4]). Therefore, ′ =β λ λn nK a q( ) ( ; ; ) and
βn ( λ ) =
( ; ) ( )
( ; ) ( )
( ; ; )
/
( )/− +
+
− +a q aq
q q a a
q K a qn
n
n
n n
n
1
1
2 1 2
1 4 λ . (2)
For the eigenvectors ξλ we have the expression
ξλ = β λn
n
n( ) 〉∑ , (3)
where βn ( λ ) is given by (2). Since the spectrum of the operator I1 is discrete, only
for a discrete set of values of λ these vectors may belong to the Hilbert space H .
This discrete set of eigenvectors determines the spectrum of I1 .
Now we look for the spectrum of I1 and for a set of polynomials, dual to
alternative q-Charlier polynomials. To this end we use the action of the operator
J : = q aqJ J− −0 0
upon the eigenvectors ξλ
, which belong to the Hilbert space H . In order to find how
this operator acts upon these vectors, one can use the q-difference equation
( ) ( )q aq Kn n
n
− − λ = – aK q K K qn n n( ) ( ) ( ) ( )λ λ λ λ λ λ+ − −− − −1 1 11 (4)
for the alternative q-Charlier polynomials Kn ( λ ) ≡ K a qn( ; ; )λ (see formula (3.22.5)
in [4]; observe that (4) can be also derived from the explicit expressions for Kn ( λ ) ) .
Multiply both sides of (4) by d nn 〉 and sum up over n, where dn are the factors in
front of K a qn( ; ; )λ in expression (2) for the βn ( λ ) . Taking into account formula (3)
and the fact that J n〉 = ( )q aq nn n− − 〉, one obtains the relation
J ξλ = – a q q
ξ λ ξ λ λ ξλ λ λ+ − −− −
−
1 1 1 1( ) . (5)
We shall see in the next section that the spectrum of the operator I1 consists of the
points qn
, n = 0, 1, 2, … . This means that J has the form of a Jacobi matrix in the
basis of eigenvectors of I1 ; that is, the pair of the operators I1 and J form an infinite
dimensional analogue of a Leonard pair (see [10] for its definition).
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
JACOBI MATRIX PAIR AND DUAL ALTERNATIVE … 617
3. Orthogonal matrix A. The aim of this section is to find, by using the operators
I1 and J, a basis in the Hilbert space H , which consists of eigenvectors of the
operator I1 in a normalized form, and to derive explicitly the unitary matrix A,
connecting this basis with the initial basis n〉 , n = 0, 1, 2, … , in H . First we have
to find the spectrum of I1 .
Let us first look at a form of the spectrum of I1 from the point of view of the spec-
tral theory of Hilbert – Schmidt operators (a detailed calculation is given below this pa-
ragraph). If λ is a spectral point of the operator I1 , then (as it is easy to see from (5))
a successive action by the operator J upon the vector ξλ (eigenvector of I1 ) leads to
the eigenvectors ξ λqm m, , , , .= ± ± …0 1 2 However, since I1 is a Hilbert – Schmidt
operator, not all of these points belong to the spectrum of I1 , since q m− λ → ∞
when m → + ∞ if λ ≠ 0. This means that the coefficient 1 − ′λ of ξ λq− ′1 in (5)
must vanish for some eigenvalue λ′
. Clearly, it vanishes when λ′ = 1. Moreover, this
is the only possibility for the coefficient of ξ λq− ′1 in (5) to vanish, that is, the point λ
= 1 is a spectral point for the operator I 1 . Let us show that the corresponding
eigenfunction ξ1 ≡ ξ
q0 belongs to the Hilbert space H .
By formula (II.6) from Appendix II in [9], one has the following equality
K a qn( ; ; )1 = 2 1 0φ ( , ; ; , )q aq q qn n− − = ( )− a qn n2
. Therefore, for the scalar product
〈 〉ξ ξ1 1, in H we have
〈 〉ξ ξ1 1, =
n
n
n
n
n n n n
a q aq
a q q a q
K a q
=
∞
+∑ − +
+0
2
1 2
21
1
1
( ; ) ( )
( )( ; )
( ; ; )( )/ =
n
n
n n
n
n n
a q aq a
q q q q=
∞
− −∑ − +
+0
2
3 1 2
1
1
( ; ) ( )
( )( ; ) ( )/ . (6)
In order to calculate this sum, we take the limit d, e → ∞ in the formula of Exercise
2.12, Chapter 2, in [9]. Since
lim ( ; ) ( ; ) ( )
,
/
d e n n
nd q e q aq de
→∞
= q aqn n n( )( )−1 ,
we obtain that the sum in (6) is equal to ( ; )− ∞aq q , that is, 〈 〉ξ ξ1 1, < ∞ and ξ1
belongs to the Hilbert space H . Thus, the point λ = 1 does belong to the spectrum
of I1 .
Let us find other spectral points of the operator I1 (recall that the spectrum of I1
is discrete). Setting λ = 1 in (5), we see that the operator J transforms ξ
q0 into a li-
near combination of the vectors ξq and ξ
q0 . Moreover, ξq belongs to the Hilbert
space H , since the series
〈 〉ξ ξq q, =
n
n
n
n
n
n n
n
a q aq
a q q a
q K q a q
=
∞
− +∑ − +
+0
2
1 2 21
1
( ; ) ( )
( )( ; )
( ; ; )( )/
is majorized by the corresponding series (6) for ξ
q0 . Therefore, ξq belongs to the
Hilbert space H and the point q is an eigenvalue of the operator I1 . Similarly, set-
ting λ = q in (5), one finds likewise that ξ
q2 is an eigenvector of I1 and the point
q2 belongs to the spectrum of I1 . Repeating this procedure, we find that all ξ
qn , n =
= 0, 1, 2, … , are eigenvectors of I1 and the set qn
, n = 0, 1, 2, … , belongs to the
spectrum of I1 . So far, we do not know yet whether other spectral points exist or not.
The vectors ξ
qn , n = 0, 1, 2, … , are linearly independent elements of the Hilbert
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
618 N. M. ATAKISHIYEV, A. U. KLIMYK
space H (since they correspond to different eigenvalues of the self-adjoint operator
I1 ). Suppose that values qn
, n = 0, 1, 2, … , constitute the whole spectrum of I1 .
Then the set of vectors ξ
qn , n = 0, 1, 2, … , is a basis in the Hilbert space H .
Introducing the notation Ξk qk:= ξ , k = 0, 1, 2, … , we find from (5) that
J Ξ k = – a q q qk
k
k
k k
kΞ Ξ Ξ+
− −
−+ − −1 11( ) .
As we see, the matrix of the operator J in the basis Ξk , k = 0, 1, 2, … , is not sym-
metric, although in the initial basis n〉 , n = 0, 1, 2, … , it was symmetric. The
reason is that the matrix ( )amn with entries a qmn m
n: ( )= β , m , n = 0, 1, 2, … ,
where βm
nq( ) are the coefficients (2) in the expansion ξ
qn = βm
n
m
q n( ) 〉∑ (see
(3)), is not unitary. This fact is equivalent to the statement that the basis Ξn = ξ
qn ,
n = 0, 1, 2, … , is not normalized. To normalize it, one has to multiply Ξn by
appropriate numbers cn (which are defined below). Let Ξ̂n = cn
Ξn
, n = 0, 1, 2, … ,
be a normalized basis. Then the matrix of the operator J is symmetric in this basis. It
follows from (5) that J has in the basis Ξ̂n{ } the form
J nΞ̂ = – c c a q c c q qn n n
n
n n n
n n
n+
−
+
−
−
− −
−+ − −1
1
1 1
1
11ˆ ˆ ( ) ˆΞ Ξ Ξ .
The symmetricity of J in the basis Ξ̂n{ } means that c c an n+
−
1
1 = c c q qn n
n n−
+
− − +−1
1
1 11( ),
that is, c cn n/ −1 = aq qn n/( )1− . Therefore, the coefficients cn are equal to
cn = c a q q qn n n
n
( )/ /
/ ( ; )+[ ]1 2 1 2
,
where c is a constant.
The expansions
ˆ ( )ξ
qn x ≡ ˆ ( )Ξn x = c q mn m
n
m
β ( ) 〉∑ ≡ â mmn
m
〉∑ (7)
interrelate two orthonormal bases in the Hilbert space H . This means that the matrix
( )âmn , m, n = 0, 1, 2, … , with entries
âmn = c qn m
nβ ( ) = c
a q
q q
a q aq
a q q a q
K q a q
n n n
n
m
m
m
m m m m
n
( )/
( )/
/
( ; )
( ; ) ( )
( )( ; )
( ; ; )
+
+
− +
+
1 2 2
1 2
1 2
1
1
, (8)
is unitary, provided that the constant c is appropriately chosen. In order to calculate
this constant, we use the relation
〈 〉ˆ , ˆΞ Ξ0 0 =
m
ma
=
∞
∑
0
0
2ˆ =
m
mc q
=
∞
∑
0
0
2 2 0β ( ) = 1.
The last sum is a multiple of the sum in (6) and, consequently, c = ( ; ) /− ∞
−aq q 1 2 .
The matrix A amn: ˆ( )= is real and orthogonal. Thus, if Ξ̂n , n = 0, 1, 2, … , is a
complete basis in H , then A A
–
1 = A
–
1
A = E, that is,
n
mn m na a∑ ′ˆ ˆ = δmm′,
m
mn mna a∑ ′ˆ ˆ = δnn′ . (9)
Substituting into the first sum over n in (9) the expressions for âmn from (8), we ob-
tain the identity
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
JACOBI MATRIX PAIR AND DUAL ALTERNATIVE … 619
n
n n n
n
m
n
m
na q
q q
K q a q K q a q
=
∞ +
′∑
0
1 2( )/
( ; )
( ; ; ) ( ; ; ) =
( ; ) ( ; )
( )
( )/−
+
∞ +
′
aq q a q q
aq
q
m m
m
m
m m
mm1 2
1 2 δ , (10)
which must yield the orthogonality relation for alternative q-Charlier polynomials. An
only gap, which remains to be clarified, is the following. We have assumed that the
points qn
, n = 0, 1, 2, … , exhaust the whole spectrum of I1. Let us show that this is
the case.
Recall that the self-adjoint operator I1 is represented by a Jacobi matrix in the
basis n〉 , n = 0, 1, 2, … . According to the theory of operators of such type (see, for
example, [2], Chapter VII), eigenvectors ξλ of I1 are expanded into series in the
basis n〉 , n = 0, 1, 2, … , with coefficients, which are polynomials in λ . These
polynomials are orthogonal with respect to some positive measure d µ ( λ ) (moreover,
for self-adjoint operators this measure is unique). The set (a subset of R ), on which
these polynomials are orthogonal, coincides with the spectrum of the operator under
discussion and this spectrum is simple.
We have found that the spectrum of I1 contains the points qn
, n = 0, 1, 2, … . If
the operator I1 had other spectral points xk , then the left-hand side of (10) would
contain other terms µx m k m kk
K x a q K x a q( ; ; ) ( ; ; )′ with positive µxk
, corresponding to
these additional points. Let us show that these additional summands do not appear. To
this end we set m = m′ = 0 in relation (10) with the additional summands. Since
K x a q0( ; ; ) = 1, we have then the equality
n
n n n
n
x
k
a q
q q k
=
∞ +
∑ ∑+
0
1 2( )/
( ; )
µ = ( ; )− ∞aq q .
According to the definition of the q-exponential function Eq ( a ) (see formula (II.2)
from Appendix II in [9]), we have
n
n n n
n
a q
q q=
∞ +
∑
0
1 2( )/
( ; )
= ( ; )− ∞aq q .
Hence, µxk k∑ = 0 and all µxk
have to vanish. This means that additional sum-
mands do not appear in (10) and hence (10) does represent the orthogonality relation
for alternative q-Charlier polynomials. Thus, the following proposition is true:
Proposition 1. The spectrum of the operator I1 coincides with the set of points
qn
, n = 0, 1, 2, … . This spectrum is simple and the vectors Ξ̂n , n = 0, 1, 2, … ,
form a complete set of eigenvectors of I1. The matrix ( )âmn with entries (8) rela-
tes the initial basis n〉{ } with the normalized basis Ξ̂n{ }.
4. Dual alternative q-Charlier polynomials. Now we consider the second iden-
tity in (9), which gives the orthogonality relation for the matrix elements âmn , consi-
dered as functions of m. Due to the expression for alternative q-Charlier polynomials
from Section 2, these functions coincide (up to multiplicative factors) with the func-
tions
F x a qn( ); | = 2 1
10φ ( , ; ; , )/x a x q qn− + , (11)
considered on the set x ∈ { }, , ,q mm− = …0 1 2 . Consequently,
âmn = c
a q
q q
aq
aq q q q a q
F q a q
n n n
n
m
m
m
m m m n
m
( )/
( )/
/
( ; )
( )
( ; ) ( ; )
( ; )|
+
∞
+
−+
−
1 2 2
1 2
1 2
1
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
620 N. M. ATAKISHIYEV, A. U. KLIMYK
and the second identity in (9) gives the orthogonality relation for F q a qn
m( ; )|− :
m
m m m
m m
m
n
m
n
maq q
a aq q q q
F q a q F q a q
=
∞ − +
∞
−
′
−∑ +
−0
2 1 21( )
( ; ) ( ; )
( ; ) ( ; )
( )/
| | =
( ; ) ( )/q q
a
qn
n
n n
nn
− +
′
1 2 δ . (12)
The functions F x a qn( ; )| can be represented in another form. Indeed, taking the
limit c → ∞ in transformation (III.8) from Appendix III in [9], one derives the rela-
tion
2 1
10φ ( , ; ; , )q aq q qm m n− +− = ( ) ( , , ; ; , )/− − − −− −a q q aq q q q am m m m n n2
3 0φ .
Therefore, we have
F q a qn
m( ; )|− = ( ) ( , , ; ; , )/− − − −− −a q q aq q q q am m m m n n2
3 0φ . (13)
The basic hypergeometric function 3 0φ in (13) is a polynomial of degree n in the va-
riable µ( ) :m q aqm m= −− , which represents a q-quadratic lattice; we denote it by
d m a qn( ( ); ; )µ : = 3 0φ ( , , ; ; , )/q aq q q q am m n n− −− − − . (14)
Then formula (12) yields the orthogonality relation
m
m m m m
m
m
n n
aq a q
aq q q q
d m d m
=
∞ −
∞
′∑ +
−0
2 3 1 21( )
( ; ) ( ; )
( ( )) ( ( ))
( )/
µ µ =
( ; ) ( )/q q
a
qn
n
n n
nn
− +
′
1 2 δ (15)
for the polynomials (14) when a > 0. As far as we know this orthogonality relation is
new. We call the polynomials d m a qn( ( ); ; )µ dual alternative q -Charlier
polynomials. Thus, we proved the following theorem.
Theorem. The polynomials d m a qn( ( ); ; )µ , given by formula (14), are ortho-
gonal on the set of points µ( ) :m q aqm m= −− , m = 0, 1, 2, … , and the orthogona-
lity relation for them is given by formula (15).
Remark. The duality of polynomials is a well-known notion in the case of
polynomials, orthogonal with respect to a finite number of points (see, for example, [9],
Chapter 7). It reflects the simple fact that a finite-dimensional matrix, orthogonal by
rows, is also orthogonal by its columns. In the case when polynomials are orthogonal
on a countable set of points, the situation is more complicated (see, for example, [11 –
13]). Usually a dual set with respect to such orthogonal polynomials is given in terms
of functions (for instance, their explicit form (11) for the case of alternative q-Charlier
polynomials is given above). One therefore needs to make one step further in order to
extract an appropriate family of dual polynomials from these functions (of course, in all
those cases when functions admit such a separation). Observe that in our approach to
the duality of q-polynomials it is not assumed that an orthogonality relation for the
initial set of polynomials is known; this orthogonality is straightforwardly derived.
Besides, one naturally extracts from a dual set of orthogonal functions an appropriate
dual family of q-polynomials. Other instances of the similar approach to the duality
can be found in [14] and [15].
Let � a
2 be the Hilbert space of functions on the set { }, , ,m = …0 1 2 with the sca-
lar product
〈 〉f f1 2, =
m
m m
m
m
m maq a
aq q q q
q f m f m
=
∞
∞
−∑ +
−0
2
3 1 2
1 2
1( )
( ; ) ( ; )
( ) ( )( )/ , (16)
where the weight function is taken from (15). The polynomials (14) are in one-to-one
correspondence with the columns of the orthogonal matrix ( )âmn and the orthogonali-
ty relation (15) is equivalent to the orthogonality of these columns. Due to (9) the
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
JACOBI MATRIX PAIR AND DUAL ALTERNATIVE … 621
columns of the matrix ( )âmn form an orthogonal basis in the Hilbert space �2 of
sequences a = { }, , ,a nn = …0 1 2 with the scalar product 〈 〉′a a, = a ann n∑ ′ . This
scalar product is equivalent to the scalar product (16) for the polynomials
d m a qn( ( ); ; )µ . The fact that the set of all columns of the matrix ( )âmn is a basis in
�
2 means that the set of all polynomials (14) forms a basis in � a
2 . Thus, the following
proposition is true:
Proposition 2. The set of polynomials d m a qn( ( ); ; )µ , n = …0 1 2, , , , form an
orthogonal basis in the Hilbert space � a
2 that is, this set is complete in � a
2 .
We have been unable to clarify yet whether the moment problem, connected with
the polynomials (14), is determinate or indeterminate. Nevertheless, Proposition 2
means that if this moment problem is indeterminate, then the measure in (15) is
extremal.
A recurrence relation for the polynomials d m a qn( ( ); ; )µ ,
( ) ( ( ))q aq d mm m
n
− − µ = – ad m q d m q q d mn
n
n
n n
n+
− −
−+ − −1 11( ( )) ( ( )) ( ) ( ( ))µ µ µ ,
where d mn( ( ))µ ≡ d m a qn( ( ); ; )µ , is readily derived from (4). A q-difference equa-
tion for d mn( ( ))µ can be obtained from the three-term recurrence relation for alterna-
tive q-Charlier polynomials K x a qn( ; ; ) .
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Received 21.01.2005
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