# Inversion formula with hypergeometric polynomials and its application to an integral equation

For any complex parameters x and ν, we provide a new class of linear inversion formulas T = A(x,ν) · S S = B(x,ν) · T between sequences S = (S_n)_n ∈N^* and T = (T_n)_n ∈N^*, where the infinite lower-triangular matrix A(x,ν) and its inverse B(x,ν) involve Hypergeometric polynomials F(·), namely { < a r r a y > . for 1 ≤ k ≤ n. Functional relations between the ordinary (resp. exponential) generating functions of the related sequences S and T are also given. These new inversion formulas have been initially motivated by the resolution of an integral equation recently appeared in the field of Queuing Theory; we apply them to the full resolution of this integral equation. Finally, matrices involving generalized Laguerre polynomials polynomials are discussed as specific cases of our general inversion scheme.

## Authors

• 10 publications
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09/19/2019

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## 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 state-of-the-art 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) ∫U−0E∗(ζR(ζ)⋅z)e−R(ζ)⋅zdζ=A(z),z∈C,

for an unknown entire function in with .

The product intervening in the argument of in (1.1) being not one-to-one 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) E∗(z)=+∞∑ℓ=1Eℓzℓℓ!,z∈C,

for an entire solution , however, drives the resolution of (1.1) to that of the infinite lower-triangular linear system

 (1.3) ∀b∈N∗,b∑ℓ=1(−1)ℓ(bℓ)Qb,ℓEℓ=Kb,

with unknown , , and coefficient matrix given by

 (1.4) Qb,ℓ=−Γ(b)Γ(1−bν)Γ(b−bν)(U−)ℓ+1x1−b1−xF(ℓ−b,−bν;−b;x),1⩽ℓ⩽b.

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 right-hand side in (1.3) is not necessary.

Diagonal coefficients , , are non-zero so that lower-triangular 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) ∀b∈N∗,b∑ℓ=1Ab,ℓ(x,ν)˜Eℓ=˜Kb,

with the reduced unknowns and right-hand side

 ˜Eℓ=(U−)ℓ+1⋅Eℓ,˜Kb=−Γ(b−bν)Γ(b)Γ(1−bν)(1−x)xb−1⋅Kb,

and coefficients

 (1.6) Ab,ℓ(x,ν)=(−1)ℓ(bℓ)F(ℓ−b,−bν;−b;x),1⩽ℓ⩽b.

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 right-hand 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 lower-triangular matrices with coefficients involving Hypergeometric polynomials; as shown in the following, other polynomial families can also be included in this pattern. In the following sub-section, we position the originality of the present contribution with respect to known inversion patterns.

### 1.2. State-of-the-art

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

 An,k=1(n−k)!n−1∏j=k(aj+k),Bn,k=ak+kan+n⋅(−1)n−k(n−k)!n∏j=k+1(aj+n)

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) An,k=n−1∏j=k(aj+bjck)n∏j=k+1(cj−ck),Bn,k=ak+bkckan+bncn⋅n∏j=k+1(aj+bjcn)n−1∏j=k(cj−cn)

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

 Ufk(z)=ck⋅Vfk(z),k∈Z,

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 multi-dimensional 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 lower-triangular 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. Lower-Triangular Systems

Let and be complex sequences such that and denote by and their respective exponential generating series, i.e.,

 (2.1) f(x)=+∞∑m=0amm!xm,g(x)=+∞∑m=0bmm!xm;

in the following, we will use the notation for the coefficient of , , in power series . For all , define the infinite lower-triangular matrices and by

 (2.2) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩An,k(x)=(−1)k(nk)n−k∑m=0(k−n)mamm!xm,Bn,k(x)=(−1)k(nk)n−k∑m=0(k−n)mbmm!xm

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) [xn−k]f(−x)g(x)=δ(n−k),1⩽k⩽n,

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

 DN(λ,μ)=N−1∑r=0(−1)rΓ(1+r−λ)Γ(1−r+μ),

we have

 (2.4) DN(λ,μ)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1μ−λ[1Γ(−λ)Γ(1+μ)−(−1)NΓ(N−λ)Γ(1−N+μ)],μ≠λsin(πλ)π[ψ(−λ)−ψ(N−λ)], μ=λ,

where denotes the logarithmic derivative .

We now proceed with the justification of Proposition 2.1.

###### Proof.

and being lower-triangular, so is their product . After definition (2.2), the coefficient , (where the latter sum over index is actually finite), of matrix reads

 Cn,k(x)= +∞∑ℓ=1(−1)ℓn!ℓ!(n−ℓ)!n−ℓ∑m=0(−1)m(n−ℓ)!am(n−ℓ−m)!m!xm× (−1)kℓ!k!(ℓ−k)!ℓ−k∑m′=0(−1)m′(ℓ−k)!bm′(ℓ−k−m′)m′!xm′

after writing for any positive integer , that is,

 (2.5) Cn,k(x)=(−1)kn!k!+∞∑ℓ=1(−1)ℓn−ℓ∑m=0(−1)mamxmm!(n−ℓ−m)!ℓ−k∑m′=0(−1)m′bm′xm′m′!(ℓ−k−m′)!.

Exchanging the summation order in (2.5) further gives

 Cn,k(x)=(−1)kn!k! ∑(m,m′)∈Δn,k(−1)mamxmm!(−1)m′bm′xm′m′!× (2.6) ∑k⩽ℓ⩽n(−1)ℓ(n−ℓ−m)!(ℓ−k−m′)!

with and where the latter summation on index equivalently reads

 ∑k⩽ℓ⩽n(−1)ℓ(n−ℓ−m)!(ℓ−k−m′)!= n−k∑r=0(−1)n−r(r−m)!(n−r−k−m′)! = (−1)nDn−k+1(m,n−k−m′)

with the index change and the notation of Lemma 2.1. The expression (2.6) for coefficient consequently reduces to

 Cn,k(x)=(−1)n+kn!k!∑(m,m′)∈Δn,k (−1)mamxmm!(−1)m′bm′xm′m′!× (2.7) Dn−k+1(m,n−k−m′)

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

 Dn−k+1(m,n−k−m′)= 1n−k−(m+m′)[1Γ(−m)Γ(1+n−k−m′)−(−1)n−k+1Γ(n−k+1−m)Γ(−m′)];

as for all non negative integers and , each fraction of the latter expression vanishes and thus

 (2.8) Dn−k+1(m,n−k−m′)=0,m+m′
• if , formula (2.4) entails

 (2.9) Dn−k+1(m,m)=limλ→msin(πλ)π[ψ(−λ)−ψ(n−k+1−λ)].

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

 limλ→msin(πλ)πψ(−λ)=0×ψ(1+m)+(−1)m=(−1)m;

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) Dn−k+1(m,m)=(−1)m,m+m′=n−k.

In view of the previous items (a) and (b), identities (2.9) and (2.10) together reduce expression (2.7) to

 Cn,k(x)= (−1)n+kn!k!n−k∑m=0(−1)mamxmm!(−1)n−k−mbn−k−mxn−k−m(n−k−m)!×(−1)m = n!k!n−k∑m=0(−1)mamxmm!bn−k−m(n−k−m)!xn−k=n!k![x]n−kf(−x)g(x)

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 lower-triangular matrices and are also given for each pair of indexes , that is,

 (2.11) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩An,k(x)=(−1)k(nk)n−k∑m=0(k−n)mam;n,km!xm,Bn,k(x)=(−1)k(nk)n−k∑m=0(k−n)mbm;n,km!xm

for . Condition (2.3) for then simply extends to

 (2.12) [xn−k]fn,k(−x)gn,k(x)=δ(n−k),1⩽k⩽n,

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 lower-triangular matrices involving Hypergeometric polynomials.

###### Theorem 2.1.

Let and define the lower-triangular matrices and by

 (2.13) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩An,k(x,ν)=(−1)k(nk)F(k−n,−nν;−n;x),Bn,k(x,ν)=(−1)k(nk)F(k−n,kν;k;x)

for . For any pair of complex sequences and , the inversion formula

 (2.14) Tn=n∑k=1An,k(x,ν)Sk⟺Sn=n∑k=1Bn,k(x,ν)Tk,n∈N∗,

holds.

###### Remark 2.1.

a) Note that the factor in the definition (2.13) of matrix is always well-defined although the third argument is a negative integer; in fact, given , write by definition ([6], 15.2.1)

 (2.15) F(k−n,−nν;−n;x)=n−k∑m=0(k−n)m(−nν)m(−n)mm!xm

and the denominator therefore never vanishes for all indexes ;

b) the polynomial factors and respectively intervening in coefficients and in definition (2.13) are deduced from each other by the substitution . This simple substitution, however, does not leave the remaining factor invariant and thus cannot carry out by itself the inversion scheme (2.14).

###### 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) F(k−n,kν,k;x)=n−k∑m=0(k−n)m(kν)m(k)mm!xm

so that , , for given with . Let and respectively denote the exponential generating function of these sequences and ; the product is then given by

 fn(−x)gk(x) =(∑m⩾0(−1)mam;nm!xm)(∑m⩾0bm;km!xm) =+∞∑m=0(−1)m(−nν)m(−n)mm!xm⋅+∞∑m=0(kν)m(k)mm!xm=∑ℓ⩾0U(n,k)ℓxℓ

where

 (2.17) U(n,k)ℓ=ℓ∑m=0(−1)m(−nν)m(−n)mm!(kν)ℓ−m(k)ℓ−m(ℓ−m)!,ℓ⩾0.

Let then ; from expression (2.17), we derive

 U(n,k)n−k= n−k∑m=0(−1)m(−nν)m(−n)mm!⋅(kν)n−k−m(k)n−k−m(n−k−m)! = n−k∑m=0(−1)mΓ(m−nν)Γ(−nν)⋅(−1)m(n−m)!n!⋅1m!⋅Γ(n−k−m+kν)Γ(kν)× Γ(k)Γ(n−k−m+k)⋅1(n−k−m)!

after writing the Pochhammer symbol for and noting that . Reducing the latter expression of gives

 U(n,k)n−k= Γ(k)n!Γ(−nν)Γ(kν)n−k∑m=0(n−m)Γ(m−nν)Γ(n−k−m+kν)m!(n−k−m)! (2.18) = Γ(k)n!Γ(−nν)Γ(kν)(X(n,k)n−k+Y(n,k)n−k)

where we introduce the sums (after decomposing )

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩X(n,k)n−k=k⋅n−k∑m=0Γ(m−nν)Γ(n−k−m+kν)m!(n−k−m)!,Y(n,k)n−k=n−k∑m=0(n−m−k)⋅Γ(m−nν)Γ(n−k−m+kν)m!(n−k−m)!.

To calculate , note that this equals to the coefficient of in the power series expansion of the product

 (+∞∑m=0Γ(m−nν)m!xm)(+∞∑m=0Γ(m+kν)m!xm)= (+∞∑m=0Γ(−nν)(−nν)mm!xm)(+∞∑m=0Γ(kν)(kν)mm!xm)=Γ(−nν)(1−x)−nν⋅Γ(kν)(1−x)kν

so that

 (2.19) X(n,k)n−k=kΓ(−nν)Γ(kν)⋅[x]n−k{(1−x)nν(1−x)kν}.

As to the sum , it equals the coefficient of in the power series expansion of the product

 (+∞∑m=0Γ(m−nν)m!xm)⋅xddx[Γ(kν)(1−x)kν]=Γ(−nν)(1−x)−nν×xΓ(kν)kν(1−x)kν+1

so that

 (2.20) Y(n,k)n−k=Γ(−nν)Γ(kν+1)⋅[x]n−k{x(1−x)nν(1−x)kν+1}.

Using formulas (2.19) and (2.20) for sums and , the expression (2.18) for then easily reduces to

 U(n,k)n−k= [x]n−kn!{Γ(k+1)(1−x)nν(1−x)kν+kνΓ(k)x(1−x)nν(1−x)kν+1} (2.21) = k!n!{[xn−k](1−x)(n−k)ν−1(1+(ν−1)x)},n⩾k.

With the series expansion , expression (2.21) for then gives

 U(n,k)n−k= k!n!{(1−(n−k)ν)n−k(n−k)!+(ν−1)(1−(n−k)ν)n−k−1(n−k−1)!} = (ν−1)Γ((n−k)(1−ν))(n−k)}

by definition of the Pochhammer symbol, and the relation applied to the argument entails

 U(n,k)n−k= (ν−1)Γ((n−k)(1−ν))(n−k)}

so that for . Now if , (2.21) reduces to

 U(n,k)n−k=[x0]{1+νx1−x}=1.

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.

###### Corollary 3.1.

For given complex parameters and , let and be sequences related by the inversion formulas (2.14) of Theorem 2.1, that is, .

Denote by and the formal ordinary generating series of and , respectively. Defining the mapping (depending on parameters and ) by

 (3.1) Ξ(z)=zz−1(1−z1−z(1−x))ν,

the relation

 (3.2) GS(z)=[1−ν1−z+ν1−z(1−x)]GT(