1. Introduction
In this Introduction, we present a general class of linear inversion formulas with coefficients involving Hypergeometric polynomials and motivate the need for such formulas. After an overview of the recent stateoftheart in the corresponding field, we summarize the main contributions of this paper.
1.1. Motivation
The need for an inversion formula whose coefficients involve Hypergeometric polynomials is motivated by the resolution of an integral equation arising from Queuing Theory [4], which can be formulated as follows:
given a constant , a real function on (with ) and an entire function in , solve the integral equation
(1.1) 
for an unknown entire function in with .
The product intervening in the argument of in (1.1) being not onetoone on interval (it vanishes at both and ), this integral equation is not amenable to a standard Fredholm equation of the first kind ([8], Chap.3, 3.1.6). An exponential power series
(1.2) 
for an entire solution , however, drives the resolution of (1.1) to that of the infinite lowertriangular linear system
(1.3) 
with unknown , , and coefficient matrix given by
(1.4) 
In (1.4), is the Euler Gamma function and denotes the Gauss Hypergeometric function with complex parameters , , ; besides, , and are known real parameters (whose specification is not needed). Recall that reduces to a polynomial with degree (resp. ) if (resp. ) equals a non positive integer; expression (1.4) for coefficient thus involves a Hypergeometric polynomial with degree in both arguments and . At this stage, the explicit expression of the righthand side in (1.3) is not necessary.
Diagonal coefficients , , are nonzero so that lowertriangular system (1.3) has a unique solution; equivalently, this proves the uniqueness of the entire solution to (1.1) with power series expansion (1.2). This solution, nevertheless, needs to be made explicit in terms of parameters; to this end, write system (1.3) equivalently as
(1.5) 
with the reduced unknowns and righthand side
and coefficients
(1.6) 
As shown in the present paper, it proves that that the linear relation (1.5) to which initial system (1.3) has been recast is always amenable to an explicit inversion for any righthand side , the inverse matrix involving also Hypergeometric polynomials. This consequently solves system (1.3) explicitly, hence integral equation (1.1).
Beside the initial motivation stemming from an integral equation, the remarkable structure of the inversion scheme obtained in this paper brings a new contribution to the realm of linear inversion formulas, namely infinite lowertriangular matrices with coefficients involving Hypergeometric polynomials; as shown in the following, other polynomial families can also be included in this pattern. In the following subsection, we position the originality of the present contribution with respect to known inversion patterns.
1.2. Stateoftheart
We here review the known classes of linear inversion formulas provided by the recent literature, most of them motivated by problems from pure Combinatorics together with the determination of remarkable relations on special functions. Given a complex sequence , it has been early shown [2] that the lower triangular matrices and with coefficients
for (with a product over an empty set being set 1) are inverses. These inversion formulas actually prove to be a particular case of the general Krattenthaler formulas [5] stating that, given complex sequences , and with for , the lower triangular matrices and with coefficients
(1.7) 
for , are inverses; the proof of (1.7) relies on the existence of linear operators , on the linear space of formal Laurent series such that
where ; the partial Laurent series , , for the inverse inverse can then be expressed in terms of the adjoint operator of . A generalization of inverse relation (1.7) to the multidimensional case when with indexes , for some integer has also been provided in [9]; as an application, the obtained relations bring summation formulas for multidimensional basic hypergeometric series.
The lower triangular matrix introduced in (1.5)(1.6), however, cannot be cast into the specific product form (1.7) for its inversion: in fact, such a product form for the coefficients of should involve the zeros , of the Hypergeometric polynomial , , in variable ; but such zeros depend on all indexes , and , which precludes the use of a factorization such as (1.7) where sequences with one index only intervene. In this paper, using functional operations on specific generating series related to its coefficients, we will show how matrix can be nevertheless inverted through a fully explicit procedure.
1.3. Paper contribution
Our main contributions can be summarized as follows:
in Section 2, we first establish an inversion criterion for a class of infinite lowertriangular matrices, which enables us to state the inversion formula for the considered class of lower triangular matrices with Hypergeometric polynomials;
in Section 3, functional relations are obtained for ordinary (resp. exponential) generating functions of sequences related by the inversion formula;
applying the latter general results, the infinite linear system (1.5) motivated above is fully solved; both the ordinary and exponential generating functions associated with its solution are, in particular, given an integral representation (Section 4.1). Finally, matrices depending on other families of special polynomials — namely, generalized Laguerre polynomials, are discussed as specific cases of our general inversion scheme (Section 4.2).
2. LowerTriangular Systems
Let and be complex sequences such that and denote by and their respective exponential generating series, i.e.,
(2.1) 
in the following, we will use the notation for the coefficient of , , in power series . For all , define the infinite lowertriangular matrices and by
(2.2) 
for (, , , denotes the Pochhammer symbol ([6], §5.2(iii)) with ). From definition (2.2), matrices and have diagonal elements equal to , , and are thus invertible.
2.1. An inversion criterion
We first state the following inversion criterion.
Proposition 2.1.
Matrices and are inverse of each other if and only if the condition
(2.3) 
on functions and holds.
The proof of Proposition 2.1 requires the following technical lemma whose proof is deferred to Appendix 5.1.
Lemma 2.1.
Let and complex numbers , . Defining
we have
(2.4) 
where denotes the logarithmic derivative .
We now proceed with the justification of Proposition 2.1.
Proof.
and being lowertriangular, so is their product . After definition (2.2), the coefficient , (where the latter sum over index is actually finite), of matrix reads
after writing for any positive integer , that is,
(2.5) 
Exchanging the summation order in (2.5) further gives
(2.6) 
with and where the latter summation on index equivalently reads
with the index change and the notation of Lemma 2.1. The expression (2.6) for coefficient consequently reduces to
(2.7) 
and we are left to calculate for all non negative and . By Lemma 2.1 applied to and , we successively derive that

if , formula (2.4) entails
as for all non negative integers and , each fraction of the latter expression vanishes and thus
(2.8) 
if , formula (2.4) entails
(2.9) We have while function has a polar singularity at every non positive integer; the limit (2.9) is therefore indeterminate () but this is solved by invoking the reflection formula , , for function ([6], Chap.5, §5.5.4). In fact, applying the latter to first gives whence
besides, the second term in (2.9) has a finite limit when since so that tends to a positive integer. From (2.9) and the latter discussion, we are left with
(2.10)
In view of the previous items (a) and (b), identities (2.9) and (2.10) together reduce expression (2.7) to
where and denote the exponential generating function of the sequence and the sequence , respectively. It follows that
is the identity matrix
if and only if condition (2.3) holds, as claimed. ∎Following the proof of Proposition 2.1, the same arguments apply to the general case when the sequences and associated with lowertriangular matrices and are also given for each pair of indexes , that is,
(2.11) 
for . Condition (2.3) for then simply extends to
(2.12) 
where (resp. ) denotes the exponential generating function of the sequence (resp. ) for given . This straightforward generalization of Proposition 2.1 will be hereafter invoked to verify the inversion criterion.
2.2. The inversion formula
We now formulate the inversion formula for lowertriangular matrices involving Hypergeometric polynomials.
Theorem 2.1.
Let and define the lowertriangular matrices and by
(2.13) 
for . For any pair of complex sequences and , the inversion formula
(2.14) 
holds.
Remark 2.1.
Proof.
To show that , it is sufficient to verify criterion (2.12). From (2.11), we first specify the associated sequences and for a given pair . On one hand, (2.15) entails , , for given and, in particular, ; similarly, write
(2.16) 
so that , , for given with . Let and respectively denote the exponential generating function of these sequences and ; the product is then given by
where
(2.17) 
Let then ; from expression (2.17), we derive
after writing the Pochhammer symbol for and noting that . Reducing the latter expression of gives
(2.18) 
where we introduce the sums (after decomposing )
To calculate , note that this equals to the coefficient of in the power series expansion of the product
so that
(2.19) 
As to the sum , it equals the coefficient of in the power series expansion of the product
so that
(2.20) 
Using formulas (2.19) and (2.20) for sums and , the expression (2.18) for then easily reduces to
(2.21) 
With the series expansion , expression (2.21) for then gives
by definition of the Pochhammer symbol, and the relation applied to the argument entails
so that for . Now if , (2.21) reduces to
The inversion condition (2.12) for is therefore fulfilled for all and we conclude that inverse relation (2.14) holds for any pair of sequences and . ∎
3. Generating functions
As a direct consequence of Theorem 2.1, remarkable functional relations can be derived for the ordinary (resp. exponential) generating functions of sequences related by the inversion formula. We first address ordinary generating functions and state the following reciprocal relations.
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