Séminaire Lotharingien de Combinatoire 81 (2020), Article B81i

HANKEL DETERMINANTS, HAHN POLYNOMIALS, AND A FORMULA OF KRATTENTHALER

DIEGO DOMINICI

Dedicated to Professor Christian Krattenthaler on the occasion of his 60th birthday

Abstract. In this note, we find a connection between an identity of C. Krattenthaler and some Hankel determinants related to the Hahn polynomials. We also consider some limiting cases related to the Meixner and Charlier polynomials.

1. Introduction

Let ℕ_{0} denote the set

ℕ_{0} = ℕ ∪ {0} = {0, 1, 2, …}.

If {μ_{n}} is a sequence of complex numbers and L : ℂ [x] → ℂ is the linear functional defined by

L[x^{n}] = μ_{n}, n ∈ ℕ_{0},

then L is called the moment functional [9] determined by the formal moment sequence {μ_{n}}. The number μ_{n} is called the moment of order n.

Suppose that {P_{n}} is a family of monic polynomials, with deg(P_{n}) = n. If the polynomials P_{n}(x) satisfy

L[P_{n}P_{m}] = h_{n}δ_{n,m}, for n, m ∈ ℕ_{0},  (1.1)

where h_{0} = μ_{0}, h_{n} ≠ 0 and δ_{n,m} is Kronecker’s delta, then {P_{n}} is called a sequence of orthogonal polynomials with respect to L. Since

L[xP_{n}P_{k}] = 0, for k ∉ {n − 1, n, n + 1} ,

the monic orthogonal polynomials P_{n}(x) satisfy the three-term recurrence relation

xP_{n}(x) = P_{n+1}(x) + β_{n}P_{n}(x) + γ_{n}P_{n−1}(x),

with initial conditions P_{−1} = 0 and P_{0} = 1, where β_{0} = \frac{μ_{1}}{μ_{0}} and

β_{n} = \frac{1}{h_{n}} L[xP_{n}^{2}] , γ_{n} = \frac{1}{h_{n−1}} L[xP_{n}P_{n−1}], for n ∈ ℕ.

Since L[xP_{n}P_{n−1}] = L [P_{n}^{2}] , we have

γ_{n} = \frac{h_{n}}{h_{n−1}} , for n ∈ ℕ,  (1.2)

and it follows that

h_{n} = μ_{0}∏_{i=1}^{n}γ_{i}.  (1.3)

2010 Mathematics Subject Classification. Primary 15A15; Secondary 33C45 33C20.

Key words and phrases. Hankel determinants, orthogonal polynomials, moments.

Let the lower triangular matrix A_{n} be defined by

(A_{n})_{i,j} = \begin{cases}a_{i,j}, & i ≥ j, \\ 0, & i < j, \end{cases} for 0 ≤ i, j ≤ n − 1,

where

x^{i} = ∑_{k=0}^{i}a_{i,k}P_{k}(x), a_{i,i} = 1.

If we define the diagonal matrix D_{n} by

(D_{n})_{i,j} = h_{i}δ_{i,j}, for 0 ≤ i, j ≤ n − 1,

and the Hankel matrix H_{n} by

(H_{n})_{i,j} = μ_{i+j}, for 0 ≤ i, j ≤ n − 1,  (1.4)

then we have the LDL^{T} factorization (see [29, Section 4.1])

H_{n} = A_{n}D_{n}A_{n}^{T}.  (1.5)

We define the corresponding Hankel determinants by Δ_{0} = 1 and

Δ_{n} = det(H_{n}), for n ∈ ℕ.

Using (1.5), we see that

Δ_{n} = ∏_{j=0}^{n−1}h_{j},  (1.6)

and using (1.3) in (1.6), we get

Δ_{n} = ∏_{j=0}^{n−1}μ_{0}∏_{i=1}^{j}γ_{i} = μ_{0}^{n}∏_{k=1}^{n−1}γ_{k}^{n−k}.  (1.7)

The identity (1.7) is sometimes called ”Heilermann formula” [39], since J. B. H. Heilermann considered the J-fraction expansion [31] ^{1}

∑_{n=0}^{∞} \frac{μ_{n}}{w^{n+1}} = \frac{μ_{0}}{w − β_{1} − \frac{γ_{1}}{w − β_{2} − \frac{γ_{2}}{w − β_{3} − \frac{γ_{3}}{...}}}},  (1.8)

in his 1845 Ph.D. thesis ”De transformatione serierum in fractiones continuas” [6, Eq. (5.2)].

Determinants have a long history and an extensive literature, see [3, 7, 47, 53, 59, 60, 61, 64], and the impressive monographs [39] and [41].

The theories of Hankel determinants and orthogonal polynomials are deeply connected, see [9, 14, 19, 27, 33, 36, 38, 42, 57].

For some applications of Hankel determinants to combinatorial problems, see [1, 10, 11, 21, 28, 30, 35, 55, 56, 62].

1We are indebted to one of the anonymous referees for pointing out this reference.

HANKEL DETERMINANTS, HAHN POLYNOMIALS, AND A FORMULA OF KRATTENTHALER 3

Some authors^{2} have computed Hankel determinants related to continuous Hahn polynomials [25], q-Hahn polynomials [8], and little q-Jacobi polynomials [32]. For extensions to continuous and discrete elliptic Selberg integrals, see [52].

In [40, Eq. (3.5)], C. Krattenthaler showed (among many other results), the identity

∑_{0≤k_{1}<k_{2}<···<k_{n}≤N}∏_{i=1}^{n} [\frac{(x)_{k_{i}}(y)_{N−k_{i}}}{k_{i}! (N − k_{i})!}] ∏_{1≤i<j≤n}(k_{j} − k_{i})^{2}

= ∏_{k=0}^{n−1} [\frac{k!}{(N − k)!}(x)_{k}(y)_{k}(x + y + k + n − 1)_{N−n+1}] , (1.9)

as a limiting case of the q-analog

∑_{0≤k_{1}<k_{2}<···<k_{n}≤N}∏_{i=1}^{n} [\frac{(x; q)_{k_{i}} (y; q)_{N−k_{i}}}{(q; q)_{k_{i}} (q; q)_{N−k_{i}}} y^{k_{i}}] ∏_{1≤i<j≤n} (q^{k_{j}} − q^{k_{i}})^{2}

= ∏_{k=0}^{n−1} [y^{k}q^{k(k−1)} \frac{(q; q)_{k}}{(q; q)_{N−k}} (x; q)_{k} (y; q)_{k} (xyq^{k+n−1}; q)_{N−n+1}] . (1.10)

He gave two different proofs of (1.10) using: 1) a Schur function identity from [44], and 2) a q-integral evaluation from [22, 37].

The purpose of this note is to give a different proof of (1.9) related to the theory of orthogonal polynomials. We also study some limiting cases, and consider some possible generalizations.

2. Main result

Suppose that the linear functional L has the form

L[p] = ∑_{k=0}^{N}c_{k}p(k), for p(x) ∈ ℂ[x],  (2.1)

for some sequence {c_{k}} . Then, the moments μ_{l} are given by

μ_{l} = ∑_{k=0}^{N}k^{l}c_{k}, for l ∈ ℕ_{0},

and the entries of the Hankel matrix (1.4) are

(H_{n})_{i,j} = μ_{i+j} = ∑_{k=0}^{N}k^{i+j}c_{k}, for 0 ≤ i, j ≤ n − 1.  (2.2)

We can obtain a representation for the determinants of H_{n}.

Proposition 1. The Hankel determinants Δ_{n} are given by

Δ_{n} = ∑_{0≤k_{1}<k_{2}<···<k_{n}≤N}(∏_{i=1}^{n}c_{k_{i}})V_{n}(k_{1}, …, k_{n}),  (2.3)

2We are indebted to one of the anonymous referee for suggesting these references.

4 DIEGO DOMINICI

where V_{n}(k_{1}, …, k_{n}) denotes the polynomial

V_{n}(k_{1}, …, k_{n}) = ∏_{1≤i<j≤n}(k_{j} − k_{i})^{2}.

Proof. If we rewrite (2.2) as

(H_{n})_{i,j} = ∑_{k=0}^{N}∑_{l=0}^{N}k^{i}c_{k}δ_{k,l}l^{j},

we see that H_{n} has the form

H_{n} = V^{T} CV,  (2.4)

where V is the (N + 1) × n Vandermonde matrix

(V)_{i,j} = i^{j}, for 0 ≤ i ≤ N, 0 ≤ j ≤ n − 1,

and C is the (N + 1) × (N + 1) diagonal matrix

(C)_{i,j} = c_{i}δ_{i,j}, for 0 ≤ i, j ≤ N.

Using the Cauchy–Binet formula [26] in (2.4), we have

det(H_{n}) = ∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} det(c_{i}δ_{i,k_{i}}) [det (k_{i}^{j})_{0≤j≤n−1}]^{2} ,

and since

[ det_{1≤i,j≤n} (k_{i}^{j−1})]^{2} = ∏_{1≤i<j≤n}(k_{j} − k_{i})^{2},

the result follows. □ Remark 2. The expression (2.3) is a particular case of Heine’s formula [34]

det_{1≤i,j≤n}(μ_{i+j}) = \frac{1}{n!}∫_{a}^{b} ∫_{a}^{b} ⋯ ∫_{a}^{b}V_{n}(x_{1}, …, x_{n})dα(x_{1})dα(x_{2}) ⋯ dα(x_{n}),

where

μ_{i} = ∫_{a}^{b}x^{i}dα(x), for i ∈ ℕ_{0}.

Heine’s formula is used extensively in random matrix theory [4, 45, 12].

In the following, we shall use standard hypergeometric notation: the Pochhammer symbol (or rising factorial) (u)_{k} [49, Eq. (5.2.4)] is defined by (u)_{0} = 1 and

(u)_{k} = u(u + 1) ⋯ (u + k − 1), for k ∈ ℕ.  (2.5)

We can also write [49, Eq. (5.2.5)]

(u)_{z} = \frac{Γ(u + z)}{Γ(u)}, if − (u + z) ∉ ℕ_{0},  (2.6)

where Γ(z) is the Gamma function. Then the generalized hypergeometric function [49, Ch. 16] is defined by

pF_{q}( \begin{matrix}a_{1}, …, a_{p} \\ b_{1}, …, b_{q} \end{matrix} ; z) = ∑_{k=0}^{∞} \frac{(a_{1})_{k} ⋯ (a_{p})_{k}}{(b_{1})_{k} ⋯ (b_{q})_{k}} \frac{z^{k}}{k!} .  (2.7)

The Stieltjes transform of the moments μ_{n} is given by

S(w) = ∑_{n=0}^{∞} \frac{μ_{n}}{w^{n+1}} = ∑_{n=0}^{∞} \frac{1}{w^{n+1}} ∑_{k=0}^{N}k^{n}c_{k} = ∑_{k=0}^{N} \frac{c_{k}}{w − k} .

If

c_{k} = \frac{(a_{1})_{k} ⋯ (a_{p})_{k}}{(b_{1})_{k} ⋯ (b_{q})_{k}} \frac{z^{k}}{k!} , for k ∈ ℕ_{0},

we have (see [17])

S(w) = \frac{1}{w} _{p+1}F_{q+1}( \begin{matrix}−w, a_{1}, …, a_{p} \\ 1 − w, b_{1}, …, b_{q} \end{matrix} ; z) .  (2.8)

If a_{1} = −N, with N ∈ ℕ, then the series (2.7) terminates, and the Stieltjes transform (2.8) is a rational function (of w). It was shown by Wall in [63, Theorem 43.1]^{3} that a rational function has the form

R(w) = ∑_{k=0}^{N} \frac{c_{k}}{w − x_{k}} , c_{k} > 0, x_{k} ∈ ℝ, x_{i} ≠ x_{j},

if and only if

R(w) = \frac{μ_{0}}{β_{1} + w − \frac{γ_{1}}{β_{2} + w − \frac{γ_{2}}{\frac{...}{β_{N−1} + w − \frac{γ_{N−1}}{β_{N} + w}}}}} ,

where β_{k} ∈ ℝ and μ_{0}, γ_{k} > 0. Thus, we recover the J-fraction expansion (1.8) and could compute the Hankel determinant Δ_{n} using Heilermann’s formula (1.7). This was a technique used (among others) by Flajolet in [23].

2.1. Hahn polynomials. Here we apply the previous considerations to the monic Hahn polynomials, which are defined by [49, Eq. (18.20.5)]

Q_{n}(x) = \frac{(α + 1)_{n} (−N)_{n}}{(n + α + β + 1)_{n}} _{3}F_{2}( \begin{matrix}−n, −x, n + α + β + 1 \\ α + 1, −N \end{matrix} ; 1) .

For α, β ∈ ℝ \ [−N, −1], the monic Hahn polynomials satisfy the orthogonality relation [16]

∑_{k=0}^{N}Q_{n}(k)Q_{m}(k)\binom{N}{k}(α + 1)_{k} (β + 1)_{N−k} = h_{n}δ_{n,m},  (2.9)

3We are indebted to one of the reviewers for pointing out this reference.

where

h_{n} = (n!)^{2}\binom{N}{n}(α + 1)_{n} (β + 1)_{n} \frac{(α + β + 2 + 2n)_{N−n}}{(α + β + 1 + n)_{n}} .  (2.10)

In order to prove our main theorem, we will need the identity

∏_{k=0}^{n} \frac{(x + 2k + 1)_{N−k}}{(x + k)_{k}} = ∏_{k=0}^{n}(x + k + n + 1)_{N−n}, for 0 ≤ n ≤ N,  (2.11)

which is easy to verify by making use of (2.6).

We now have all the elements necessary to show our main result.

Theorem 3. For all 1 ≤ n ≤ N and α, β ∈ ℝ \ [−N, −1], we have

∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} ∏_{i=1}^{n} [\binom{N}{k_{i}}(α + 1)_{k_{i}} (β + 1)_{N−k_{i}}] V_{n}(k_{1}, …, k_{n})

= ∏_{k=0}^{n−1} [(k!)^{2}\binom{N}{k}(α + 1)_{k} (β + 1)_{k} (α + β + 1 + k + n)_{N−n+1}] . (2.12)

Proof. Let

c_{k} = \binom{N}{k}(α + 1)_{k} (β + 1)_{N−k}.

From (2.9), we see that the monic orthogonal polynomials associated with the linear functional L defined by (2.1) are the Hahn polynomials.

On the other hand, we have from (2.3)

det_{0≤i,j≤n−1} (L[x^{i+j}]) = ∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} (∏_{i=1}^{n}c_{k_{i}})V_{n}(k_{1}, …, k_{n}).

Using (1.6) and (2.10), we get

∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} ∏_{i=1}^{n} [\binom{N}{k_{i}}(α + 1)_{k_{i}} (β + 1)_{N−k_{i}}] V_{n}(k_{1}, …, k_{n})

= ∏_{k=0}^{n−1} [(k!)^{2}\binom{N}{k}(α + 1)_{k} (β + 1)_{k} \frac{(α + β + 2 + 2k)_{N−k}}{(α + β + 1 + k)_{k}} ] .

If we use (2.11), with x = α + β + 1, we have

∏_{k=0}^{n−1} \frac{(α + β + 2 + 2k)_{N−k}}{(α + β + 1 + k)_{k}} = ∏_{k=0}^{n−1}(α + β + 1 + k + n)_{N−n+1},

and the result follows. □

Corollary 4. For all 1 ≤ n ≤ N, we have

∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} ∏_{i=1}^{n} [\frac{(x)_{k_{i}} (y)_{N−k_{i}}}{k_{i}! (N − k_{i})!} ] V_{n}(k_{1}, …, k_{n})

= ∏_{k=0}^{n−1} [\frac{k!}{(N − k)!}(x)_{k} (y)_{k} (x + y + k + n − 1)_{N−n+1}] . (2.13)

Proof. If we set α = x − 1, β = y − 1 in (2.12) and divide both sides of by (N!)^{n}, we obtain (2.13). To remove any restrictions on x, y, we observe that (2.13) is an identity between polynomials in x and y of degree

∑_{k=0}^{n−1}(k + k + N − n + 1) = Nn.

According to Theorem 3, Equation (2.13) is true for x, y ∉ [−N + 1, 0], and therefore it is true for all x, y. □

2.2. Meixner polynomials.

Lemma 5. Let 0 ≤ k ≤ n ≤ N and w > 0. Then, as N → ∞, we have

\frac{N!}{(N − k)!} \frac{(1 + wN)_{N−k}}{(1 + wN)_{N}} ∼ (w + 1)^{−k},  (2.14)

and, for all a > 0,

\frac{N!}{(N − k)!} \frac{(1 + wN)_{k}}{(1 + wN)_{N}} (a + k + n + wN)_{N−n+1} ∼ w^{1−a−n}(w + 1)^{a+k}N^{2k−n+1}. (2.15)

Proof. From (2.6), we have

\frac{N!}{(N − k)!} \frac{(1 + wN)_{N−k}}{(1 + wN)_{N}} = \frac{Γ(N + 1)}{Γ(N + 1 − k)} \frac{Γ(Nw + N + 1 − k)}{Γ(Nw + N + 1)}.

Using Stirling’s formula [49, Eq. (5.11.1)]

ln Γ(z) = (z − \frac{1}{2}) ln(z) − z + \frac{1}{2} ln(2π) + O (z^{−1}) , for z → ∞,  (2.16)

we obtain

ln [\frac{Γ(N + 1)}{Γ(N + 1 − k)} \frac{Γ(Nw + N + 1 − k)}{Γ(Nw + N + 1)}] = −k ln(w + 1) + O (N^{−1}) , for N → ∞.

Similarly, from (2.6) we have

\frac{N!}{(N − k)!} \frac{(1 + wN)_{k}}{(1 + wN)_{N}} (a + k + n + wN)_{N−n+1}

= \frac{Γ(N + 1)}{Γ(N + 1 − k)} \frac{Γ(Nw + k + 1)}{Γ(Nw + N + 1)} \frac{Γ(Nw + N + k + a + 1)}{Γ(Nw + n + k + a)}.

Then, using (2.16), we get

ln [\frac{Γ(N + 1)}{Γ(N + 1 − k)} \frac{Γ(Nw + k + 1)}{Γ(Nw + N + 1)} \frac{Γ(Nw + N + k + a + 1)}{Γ(Nw + n + k + a)}]

= (2k − n + 1) ln(N) + (a + k) ln(w + 1) − (a + n − 1) ln(w) + O (N^{−1})

as N → ∞. □

Corollary 6. Let 0 < z < 1, a > 0 and n = 1, 2, …. Then

∑_{0≤k_{1}<k_{2}<···<k_{n}}∏_{i=1}^{n} [\frac{(a)_{k_{i}} z^{k_{i}}}{k_{i}!} ] V_{n}(k_{1}, …, k_{n}) = ∏_{k=0}^{n−1} [k!(a)_{k} z^{k} (1 − z)^{−a−2k}] . (2.17)

8 DIEGO DOMINICI

Proof. Multiplying both sides of (2.13) by

[\frac{N!}{(y)_{N}} ]^{n} ,

setting

x = a, y = 1 + \frac{1 − z}{z} N,

and using (2.14)–(2.15), we obtain

∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} ∏_{i=1}^{n} [\frac{N!}{(y)_{N}} \frac{(a)_{k_{i}}(y)_{N−k_{i}}}{k_{i}! (N − k_{i})!}] V_{n}(k_{1}, …, k_{n})

∼ ∑_{0≤k_{1}<k_{2}<···<k_{n}} ∏_{i=1}^{n} [\frac{(a)_{k_{i}} z^{k_{i}}}{k_{i}!} ] V_{n}(k_{1}, …, k_{n}), for N → ∞,

and

∏_{k=0}^{n−1} [\frac{N!}{(y)_{N}} \frac{k!}{(N − k)!}(a)_{k} (y)_{k} (a + y + k + n − 1)_{N−n+1}]

∼ ∏_{k=0}^{n−1} [k! (a)_{k} (\frac{1 − z}{z} )^{1−a−n} z^{−a−k}N^{2k−n+1}] , for N → ∞.

However,

∏_{k=0}^{n−1}(1 − z)^{1−a−n} = (1 − z)^{−n(a+n−1)} = ∏_{k=0}^{n−1} (1 − z)^{−a−2k},
∏_{k=0}^{n−1}z^{n−1−k} = z^{\frac{1}{2} n(n−1)} = ∏_{k=0}^{n−1}z^{k},

and

∏_{k=0}^{n−1}N^{2k−n+1} = 1.

Therefore,

lim_{N→∞}  ∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} ∏_{i=1}^{n} [\frac{(a)_{k_{i}} z^{k_{i}}}{k_{i}!} ] V_{n}(k_{1}, …, k_{n}) = ∏_{k=0}^{n−1} [k! (a)_{k} z^{k} (1 − z)^{−a−2k}] . □

The monic Meixner polynomials are defined by [49, Eq. (18.20.7)]

M_{n}(x) = (a)_{n} (1 − z^{−1})^{−n} _{2}F_{1}[ \begin{matrix}−n, −x \\ a \end{matrix} ; 1 − z^{−1}] .

For a > 0 and 0 < z < 1, the monic Meixner polynomials satisfy the orthogonality relation [16]

∑_{k=0}^{∞}M_{n}(k)M_{m}(k)\frac{(a)_{k}}{k!} z^{k} = n! (a)_{n} z^{n}(1 − z)^{−a−2n}δ_{n,m}.  (2.18)

If we choose

c_{k} = \frac{(a)_{k}}{k!} z^{k}, for k ∈ ℕ_{0},  (2.19)

we see from (2.18) that the monic orthogonal polynomials associated with the linear functional L defined by

L[p] = lim_{N→∞}∑_{k=0}^{N}c_{k}p(k)  (2.20)

are the Meixner polynomials. Now, from (2.3), we have

det_{0≤i,j≤n−1} (L[x^{i+j}]) = lim_{N→∞}  ∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} (∏_{i=1}^{n}c_{k_{i}})V_{n}(k_{1}, …, k_{n}),  (2.21)

and using (1.6) and (2.18), we get

lim_{N→∞} ∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} ∏_{i=1}^{n} [\frac{(a)_{k_{i}} z^{k_{i}}}{k_{i}!} ] V_{n}(k_{1}, …, k_{n}) = ∏_{k=0}^{n−1} [k! (a)_{k} z^{k}(1 − z)^{−a−2k}] ,

in agreement with (2.17).

Remark 7. The moments associated with (2.19) are given by [17]

μ_{n}(z) = ∑_{k=0}^{∞}k^{n} (a)_{k} \frac{z^{k}}{k!} = (1 − z)^{−a−n}P_{n}(z),

where P_{n}(z) is the polynomial

P_{n}(z) = ∑_{k=0}^{n}S(n, k) (a)_{k} z^{k} (1 − z)^{n−k},  (2.22)

and the coefficients S(n, k) are the Stirling numbers of the second kind defined by [49, Eq. (26.8)]

S(n, k) = \frac{1}{k!}∑_{j=0}^{k}\binom{k}{j}(−1)^{k−j}j^{n}.

Since

Δ_{n} = (1 − z)^{−an− \frac{1}{2} n(n−1)} det_{0≤i,j≤n−1}(P_{i+j}(z)),

we see from (2.17) that

det_{0≤i,j≤n−1}(P_{i+j}(z)) = z^{\frac{1}{2} n(n−1)}∏_{k=0}^{n−1}k! (a)_{k}.  (2.23)

The polynomials (2.22) and their Hankel determinants (2.23) seem not to have been studied before.

2.3. Charlier polynomials.

Corollary 8. Let z > 0 and n = 1, 2, …. Then

∑_{0≤k_{1}<k_{2}<···<k_{n}}∏_{i=1}^{n} [\frac{z^{k_{i}}}{k_{i}!} ] V_{n}(k_{1}, …, k_{n}) = ∏_{k=0}^{n−1} [(k!)^{2} \frac{z^{k}}{k!} e^{z}] .  (2.24)

Proof. If we do the replacement

z → \frac{z}{z + a}

in (2.17), we obtain

∑_{0≤k_{1}<k_{2}<···<k_{n}}∏_{i=1}^{n} [\frac{(a)_{k_{i}} (\frac{z}{z+a})^{k_{i}}}{k_{i}!} ] V_{n}(k_{1}, …, k_{n})

= ∏_{k=0}^{n−1} [(k!)^{2} \frac{(a)_{k} (\frac{z}{z+a})^{k}}{k!} (1 − \frac{z}{z + a})^{−a−2k}] .

The result follows from the limits

lim_{a→∞} \frac{(a)_{k}}{(z + a)^{k}} = lim_{a→∞}∏_{j=0}^{k−1} \frac{a + j}{a + z} = 1,

and

lim_{a→∞} (1 − \frac{z}{z + a})^{−a−2k} lim_{a→∞} (1 + \frac{z}{a})^{a+2k} = e^{z}.  □

The monic Charlier polynomials are defined by [49, Eq. (18.20.7)]

C_{n}(x) = (−z)^{n} _{2}F_{0}[ \begin{matrix}−n, −x \\ − \end{matrix} ; −z^{−1}] .

For z > 0, the monic Charlier polynomials satisfy the orthogonality relation [49, Eq. (18.19.1)]

∑_{k=0}^{∞}C_{n}(k)C_{m}(k)\frac{z^{k}}{k!} = n!z^{n}e^{z}δ_{n,m}.  (2.25)

If we choose

c_{k} = \frac{z^{k}}{k!} , for k ∈ ℕ_{0},  (2.26)

we see from (2.25) that the monic orthogonal polynomials associated with the linear functional L defined by (2.20) are the Charlier polynomials. Using (1.6), (2.3) and (2.25), we get

lim_{N→∞} ∑_{0≤k_{1}<k_{2}<···<k_{n}≤N} ∏_{i=1}^{n} [\frac{z^{k_{i}}}{k_{i}!} ] V_{n}(k_{1}, …, k_{n})^{2} = ∏_{k=0}^{n−1} (k!z^{k}e^{z}) ,

in agreement with (2.24).

Remark 9. The moments associated with (2.26) are given by [17]

μ_{n}(z) = ∑_{k=0}^{∞}k^{n} \frac{z^{k}}{k!} = e^{z}T_{n}(z),

where T_{n}(z) is the Touchard (or exponential, or Bell) polynomial

T_{n}(z) = ∑_{k=0}^{n}S(n, k)z^{k}.

We clearly have

Δ_{n} = e^{nz} det_{0≤i,j≤n−1}(T_{i+j}).

The determinant

det_{0≤i,j≤n−1}(T_{i+j}) = z^{\frac{1}{2} n(n−1)}∏_{k=0}^{n−1}k!

has been computed by several authors in many different ways, see [5, 20, 24, 36, 46, 50, 51, 54]. The special case z = 1 (Bell numbers), was considered in [2, 13, 43, 58, 65].

3. Conclusions

We have established a connection between C. Krattenthaler’s identity (1.9) and the Hankel determinants of moments of Hahn polynomials. As we mentioned at the end of the last section, the corresponding identity for Hankel determinants of Charlier polynomials has appeared in the literature multiple times.

We have not been able to find any other instance of the determinants

det_{0≤i,j≤n−1} (∑_{k=0}^{N}k^{i+j}c_{k}) , det_{0≤i,j≤n−1} (∑_{k=0}^{∞}k^{i+j}c_{k}) ,

for general c_{k}, or at least for c_{k} being a hypergeometric term (we do not claim that they do not exist, but we have not uncovered a single reference). That is why we were so amazed to learn about (1.9).

The next case of interest will be

c_{k} = \frac{(α_{1})_{k} (α_{2})_{k} (α_{3})_{k}}{(β_{1})_{k} (β_{2})_{k}} \frac{1}{k!}, for k ∈ ℕ_{0},

which is the weight function for the Generalized Hahn polynomials of type II introduced in [18].

In his work [40], Krattenthaler employed Schur functions to establish (1.9). In [15], we used Schur functions to compute the Hankel determinants of the Meixner and Charlier polynomials, and consider some extensions.

Acknowledgement

This work was done while visiting the Johannes Kepler Universität Linz and supported by the strategic program ”Innovatives OÖ – 2010 plus” from the Upper Austrian Government.

I thank George Andrews, Christian Krattenthaler and Veronika Pillwein for their helpful suggestions and encouragement.

Finally, I also wish to express our gratitude to the anonymous referees, who provided us with invaluable suggestions and comments that greatly improved our first draft of the paper.

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Johannes Kepler University Linz, Doktoratskolleg “Computational Mathematics”, Altenberger Straße 69, 4040 Linz, Austria

Permanent address: Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr., New Paltz, NY 12561-2443, USA

  Email address: diego.dominici@dk-compmath.jku.at