WEIGHTED DERANGEMENTS AND THE LINEARIZATION

COEFFICIENTS OF ORTHOGONAL SHEFFER POLYNOMIALS

Jiang Zeng

Abstract. The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials. In particular, we show that Viennot’s combinatorial interpretations of the moments can be derived directly from their classical analytical expressions and that the linearization coefficients of Meixner-Pollaczek polynomials have an analogous interpretation in the model of derangements as those of Laguerre and Meixner polynomials.

1. Introduction

Let {p_{n}(x) : n = 0, 1, 2, … } be a sequence of polynomials, orthogonal on the real line with respect to a positive measure dα(x). One of the questions concerning the polynomials p_{n}(x) is to determine, or at least say something useful about, the coefficients a(k, n, m) in the expansion of

(1.0)  p_{n}(x)p_{m}(x) = ∑_{k=0}^{n+m} a(k, n, m) p_{k}(x).

This is usually called the linerization problem in the literature [4]. Taking orthogonality into account, we have then the equivalent equation :

∫_{−∞}^{+∞} p_{n}(x)p_{m}(x)p_{k}(x)dα(x) = a(k, n, m) ∫_{−∞}^{+∞} p_{k}(x)p_{k}(x)dα(x).

The problem thus turns out to deal with the integrals of a product of three polynomials. More generally, we can define the linearization coefficients of p_{n}(x), abbreviated by l.c. in what follows, to be the following integral of a product of m arbitrary polynomials :

(1.1)  ∫_{−∞}^{+∞} ∏_{i=1}^{m} p_{n_{i}}(x)dα(x).

1991 Mathematics Subject Classification. A. M. S. (1980) subject classification : 05A15, 33A65.

Key words and phrases. Sheffer, Hermite, Charlier, Laguerre, Meixner, Meixner-Pollaczek polynomials, linearization coefficients, moments, derangement, permutation, partition, excedance, maximum..

Any such integral can, of course, be evaluated in finite terms by inserting the polynomial expressions for p_{n_{i}}(x) and integrating term by term. However, as pointed by several authors [4, 20, 37], the interesting problem is not just any representation of (1.1) but the one in which the nonnegativity of the integral is apparent for some range of values of the parameters. Although it is well-known that the integral (1.1) is positive for most classical orthogonal polynomials [3], there is generally no simple expression for (1.1). In the past two decades, many authors also considered the refined problem of finding combinatorial interpretations of (1.1) for the classical orthogonal polynomials (See, for example, [6, 7, 8, 14, 17, 20, 29, 22, 40, 41]). In view of their analytical aspects, it should be interesting to find a geometric setup of which the generating polynomial with positive weight is the integral (1.1) and which reduces to orthogonality when m = 2. The main object of the present paper is to show that such a combinatorial interpretation does exist for all the orthogonal Sheffer polynomials.

Recall that a sequence of polynomials p_{1}(x), p_{2}(x), … is said to be Sheffer type if and only if its exponential generating function has the form :

(1.2)  ∑_{n≥0} p_{n}(x)\frac{t^{n}}{n!} = f(t) exp(xg(t)),

where f(t), g(t) are formal power series with f(0) = 1, g(0) = 0 and g′(0) ≠ 0. As proved by Meixner [12, 34], there are only five classes of orthogonal Sheffer polynomials, i.e., the Hermite, Laguerre, Charlier, Meixner and Meixner-Pollaczek polynomials. To be more precise, we now give the explicit generating functions and weight functions of these polynomials. Throughout this paper, we denote by (x)_{n} the shifted factorial (x)_{n} = x(x + 1) ⋯ (x + n − 1).

(a) Hermite polynomials (H_{n}(x))_{n≥0} :

(1.3)  ∑_{n≥0} H_{n}(x)\frac{t^{n}}{n!} = exp (2xt − t^{2}),

orthogonal on ℝ with respect to the weight function x ↦ e^{−x^{2}}.

(b) Charlier polynomials (C_{n}(x; a))_{n≥0} :

(1.4)  ∑_{n≥0} C_{n}(x; a)\frac{t^{n}}{n!} = e^{t}(1 − \frac{t}{a})^{x},

orthogonal on ℤ_{+} with respect to the weight function x ↦ a^{x}/x!.

(c) Laguerre polynomials (L_{n}^{(α)}(x))_{n≥0} :

(1.5)  ∑_{n≥0} L_{n}^{(α)}(x)\frac{t^{n}}{n!} = (1 − t)^{−α−1} exp(\frac{xt}{1 − t}),

orthogonal on ℝ_{+} with respect to the weight function x ↦ x^{α}e^{−x}.

(d) Meixner polynomials (M_{n}(x; β, c))_{n≥0} :

(1.6)  ∑_{n≥0} M_{n}(x; β, c)\frac{t^{n}}{n!} = (1 − \frac{t}{c})^{x}(1 − t)^{−x−β},

WEIGHTED DERANGEMENTS AND THE LINEARIZATION COEFFICIENTS OF ORTHOGONAL SHEFFER POLYNOMIALS3

orthogonal on ℤ_{+} with respect to the weight function x ↦ (β)_{x}c^{x}/x!.

(e) Meixner-Pollaczek polynomials (P_{n}(x; δ, η))_{n≥0} :

(1.7)  ∑_{n≥0} P_{n}(x; δ, η)\frac{t^{n}}{n!} = [(1 + δt)^{2} + t^{2}]^{−η/2} exp[x arctan(\frac{t}{1 + δt})],

orthogonal on ℝ with respect to the weight function ( here a misprint in [12, 40] has been corrected.)

x ↦ [Γ(η/2)]^{−2}|Γ((η + ix)/2)|^{2} exp(x arctan δ).

For a fixed positive integer m, let n be a sequence n = (n_{1}, …, n_{m}) of m nonnegative integers. The linearization coefficients of the five sets of the Sheffer polynomials can then be defined, with appropriate normalizations, as follows.

(1.8)  H(n) = \frac{1}{\sqrt{π}} ∫_{−∞}^{+∞} (∏_{i=1}^{m} 2^{−n_{i}/2}H_{n_{i}}(x)) e^{−x^{2}}dx.

(1.9)  C(n; a) = e^{−a} ∑_{x=0}^{∞} (∏_{i=1}^{m}(−a)^{n_{i}}C_{n_{i}}(x; a)) \frac{a^{x}}{x!} .

(1.10)  L(n; α) = \frac{1}{Γ(α + 1)} ∫_{0}^{+∞} (∏_{i=1}^{m}(−1)^{n_{i}}L_{n_{i}}^{(α)}(x)) x^{α}e^{−x}dx.

(1.11)  M(n; β, c) = (1 − c)^{β} ∑_{x=0}^{∞} (∏_{i=1}^{m}(−c)^{n_{i}} M_{n_{i}}(x; β, c)) \frac{c^{x}(β)_{x}}{x!} .

(1.12)  P(n; δ, η) = \frac{1}{∫_{−∞}^{+∞} w(x)dx} ∫_{−∞}^{+∞} (∏_{i=1}^{m} P_{n_{i}}(x; δ, η)) w(x)dx,

where w(x) = [Γ(η/2)]^{−2}|Γ((η + ix)/2)|^{2} exp(x arctan δ).

Although the combinatorial aspects of the l.c. of simple Laguerre polynomials L_{n}^{(0)}(x) were first studied (cf. [17, 6, 29, 10]), the interpretation of the l.c. of the general Laguerre polynomials was found only recently by Foata and Zeilberger [20]. The interpretation of the l.c. of Hermite polynomials is due to Godsil [26] and Azor, Gillis and Victor [8]. The combinatorial interpretations of the l.c of Charlier and Meixner polynomials were recently found by Zeng [41]. Thus, as far as the linerization problems of Sheffer polynomials are concerned, there is still one interesting question remaining : Is there an analogous interpretation for the linearization coefficients of Meixner-Pollaczek polynomials ? The object of this paper is to give an affirmative answer to this question, but we also present a new approach to the linearization problem, which is based on generating functions and the interpretation of the moments.

Various approaches have been used hitherto to investigate the combinatorial interpretations of the linearization coefficients. In the earliest paper [17], Even and Gillis showed, by identifying the recurrence relations, that the l.c. of the simple Laguerre polynomials counts the “generalized derangements” : rearrangements of a multiset of objects of different “colours” with the property that each object goes to an object of a different colour. Shortly after this, Askey and Ismail [6] observed that this result can be derived more directly by identifying the generating function of the integral (1.10) (α = 0) with that of the number of derangements, which was evaluated by MacMahon’s Master theorem. Similarly, by refining the counting of derangements, they also found a combinatorial interpretation of the l.c. of Meixner polynomials M_{n}(x; c, β) for β = 1 [6]. In addition, Carlitz [10] also gave a short proof of Even and Gillis’ result by carrying out the integral (1.10) explicitly, always in α = 0 case, and then applying an inversion formula or the principle of inclusionexclusion. One decisive step was made by Foata and Zeilberger [20] in order to interpret the general α of (1.10). Instead of multiset, they consider the permutations of a total ordered “colored set”, which permits them to introduce the notion of cycle. Based on this geometric setup, Foata and Zeilberger [20] and Zeng [41] successfully generalized both Askey-Ismail and Carlitz’ methods to figure out the interpretations of the l.c. of Laguerre, Meixner, Charlier and Krawchouk polynomials, by using especially the exponential formula and the β-extension of the MacMahon Master theorem.

Another different approach was initiated by Jackson [29]. He observed that the result of Even and Gillis [17] can be naturally derived from the interpretation of the moments and from the rook polynomial interpretations of the simple Laguerre polynomials [38]. This approach has the advantage of relating naturally the three different objects : the (rook) polynomials, the moments, and the linearization coefficients. Similarly, Godsil [26] has given an interpretation of the l.c. of Hermite polynomials, but instead of rook polynomials he used matching polynomials. This method has later been further developed by De Sainte-Catherine and Viennot [14] to provide an interpretation of the l.c. of Tchebycheff polynomials of the second kind; they also found the explicit killing involution under the principle of inclusionexclusion involved in the three above proofs. More recently, Gessel [22] produced a common generalization of rook and matching polynomials to give, among others, new proofs of the known interpretations of the l.c. of the Hermite, general Laguerre, and Charlier polynomials.

Our interpretation of the l.c. of the Meixner-Pollaczek polynomials is again based on the model of derangements. Indeed, we will show that integral (1.12) counts the derangements according to the three statistics : excedances, decedance and cycles (cf. Theorem 4). This weight was originally inspired by Viennot’s combinatorial interpretations for the moments of the Sheffer polynomials [40]. Moreover, we will propose two new approaches to the interpretation of the linearisationa coefficients of Sheffer polynomials based on a segment decomposition of derangements.

For other combinatorial and group theoretical aspects of the orthogonal polynomials discussed in this paper, we refer the reader to [9, 19, 31, 32, 39, 40].

This paper is organized as follows. In § 2, we first evaluate the generating functions of the moments of Sheffer polynomials and then give their combinatorial interpretations (cf. Theorem 1), which are essentially the combinatorial definitions of the moments of Sheffer polynomials due to Viennot [40]. In § 3, we use the generating functions of the moments to evaluate the generating functions of their l.c., in particular, that of the l.c. of Meixner-Pollaczek polynomials (cf. Theorem 2). In § 4, we introduce the notion of c-derangements and study their properties. In § 5, we evaluate the generating functions of the c-derangements according to various weights (cf. Theorem 3). In § 6, we deduce the combinatorial interpretations of the l.c. of Sheffer polynomials from their generating functions (cf. Theorem 4) and show that in the case where δ = 0 it has another interesting interpretation in the context of general up-down permutations (cf. Theorem 5).

2. The moments and combinatorial preliminaries

It seems convenient to consider the integral in (1.1) as a formal linear functional on the vector space of the polynomials. Thus the functionals corresponding to the Hermite, Charlier, Laguerre, Meixner, and Meixner-Pollaczek polynomials can be defined by their values on the base {x^{n}}_{n≥0} as follows.

(2.1)  ν(x^{n}) = \frac{1}{\sqrt{π}} ∫_{−∞}^{+∞}(\sqrt{2}x)^{n} e^{−x^{2}}dx.

(2.2)  μ(x^{n}) = e^{−a} ∑_{x=0}^{∞} x^{n} \frac{a^{x}}{x!} .

(2.3)  ψ(x^{n}) = \frac{1}{Γ(α + 1)} ∫_{0}^{+∞} x^{n} x^{α}e^{−x}dx.
(2.4)  ρ(x^{n}) = (1 − c)^{β} ∑_{x=0}^{∞} x^{n} \frac{c^{x}(β)_{x}}{x!} .

(2.5)  φ(x^{n}) = \frac{1}{∫_{−∞}^{+∞} w(x)dx} ∫_{−∞}^{+∞} x^{n} w(x)dx,

where w(x) = [Γ(η/2)]^{−2}|Γ((η + ix)/2)|^{2} exp(x arctan δ). One notices that the above quantities are actually the moments of the corresponding polynomials.

Lemma 1. We have

(2.6)  ∫_{−∞}^{∞} x^{n} e^{−(π−2ϕ)x}|Γ(a + ix)|^{2} dx = 2^{−n} π Γ(2a)\frac{d^{n}}{dϕ^{n}} (2 sin ϕ)^{−2a}.

Proof. According to Pollaczek [35], we have

∫_{−∞}^{∞} e^{−(π−2ϕ)x}|Γ(a + ix)|^{2} dx = π Γ(2a)(2 sin ϕ)^{−2a}.

Differentiating the above formula n times and then dividing both sides by 2^{n}, we obtain immediately (2.6).

Proposition 1. The following identities hold.

(2.7)  ∑_{n≥0} ν(x^{n})\frac{t^{n}}{n!} = exp(t^{2}/2).

(2.8)  ∑_{n≥0} μ(x^{n})\frac{t^{n}}{n!} = exp(a(e^{t} − 1)).

(2.9)  ∑_{n≥0} ψ(x^{n})\frac{t^{n}}{n!} = (1 − t)^{−α−1}.

(2.10)  ∑_{n≥0} ρ(x^{n})\frac{t^{n}}{n!} = (\frac{1 − c}{1 − ce^{t}} )^{β}.

(2.11)  ∑_{n≥0} φ(x^{n})\frac{t^{n}}{n!} = (cos t − δ sin t)^{−η}.

Proof. Formulae (2.8) and (2.10) can be derived directly by inserting the series expressions and exchanging the order of the summations. Next, we notice that ν(x^{2n+1}) = 0 and

ν(x^{2n}) = \frac{1}{\sqrt{π}} ∫_{−∞}^{+∞}(\sqrt{2} x)^{2n} e^{−x^{2}}dx = 1 · 3 ⋯ (2n − 1),
ψ(x^{n}) = \frac{1}{Γ(α + 1)} ∫_{0}^{+∞} x^{n} x^{α}e^{−x}dx = (α + 1)_{n}.

Formulae (2.7) and (2.9) then follow immediately. Finally, we will give two proofs of (2.11). The first one uses only the special case n = 0 of Lemma 1. By definition (2.5), we have

∑_{n=0}^{∞} φ(x^{n})\frac{t^{n}}{n!} = \frac{∫_{−∞}^{+∞} e^{xt} w(x)dx}{∫_{−∞}^{+∞} w(x)dx}

= \frac{∫_{−∞}^{+∞}|Γ((η + ix)/2)|^{2}e^{x(t+arctan δ)}dx}{∫_{−∞}^{+∞}|Γ((η + ix)/2)|^{2}e^{x arctan δ}dx}

= \frac{∫_{−∞}^{+∞}|Γ(η/2 + iy)|^{2}e^{y(2t+2 arctan δ)}dy}{∫_{−∞}^{+∞}|Γ(η/2 + iy)|^{2}e^{2y arctan δ}dy}

= [\frac{sin(π/2 + t + arctan δ)}{sin(π/2 + arctan δ)}]^{−η}

= (cos t − δ sin t)^{−η}.

The second one uses Lemma 1 and Lagrange’s inversion formula. Comparing definition (2.5) and Lemma 1, we see that φ(x^{n}) is the evaluation of the following function

(sin ϕ)^{η} \frac{d^{n}}{dϕ^{n}} (sin ϕ)^{−η}

at ϕ = \frac{π}{2} +arctan δ. Thanks to Lagrange’s inversion formula or just Taylor formula [36], we have

∑_{n=0}^{∞}(sin ϕ)^{η} \frac{d^{n}}{dϕ^{n}} (sin ϕ)^{−η} \frac{t^{n}}{n!} = (sin ϕ)^{η}[sin(ϕ + t)]^{−η}

= (cos t + cot ϕ sin t)^{−η}.

Substituting ϕ = \frac{π}{2} + arctan δ into the last line, we obtain (2.11).

In order to give combinatorial interpretations of these moments, we first need to introduce some basic definitions and notations. Let ℕ be the set of nonnegative integers and let [n] = {1, …, n} for n ∈ ℕ, by convention [0] = ∅. Let 𝔖_{n} be the set of permutations of [n]. For σ ∈ 𝔖_{n}, we say that σ has an

excedance at i ∈ [n] if σ(i) > i,

decedance at i ∈ [n] if σ(i) < i,

maximun at i ∈ [n] if σ^{−1}(i) < i > σ(i), and denote by exc σ, dec σ, and max σ respectively the numbers of excedances, decedance and maxima of σ. A cycle of σ is a sequence (a, σ(a), …, σ^{k−1}(a)) such that σ^{k}(a) = a and σ^{l}(a) ≠ a for 1 < l < k. The length of this cycle is k. In case k = 1, we say that a is a fixed point of σ. Clearly, each permutation σ can be factorized as a product of disjoint cycles. We denote by cyc σ the number of the cycles of σ. A permutation σ is said to be circular if cyc π = 1. Furthermore, we call σ a derangement of [n] if σ(i) ≠ i for all i ∈ [n] or equivalently all the cycles of σ have lengths ≥ 2 and denote by 𝔇_{n} the set of derangements of [n]. Finally, we call σ an involution if σ^{2} = id. and denote by ℑ_{n} the set of fixed point free involutions on [n].

For a finite set S, we denote by |S| the cardinal of S. A partition of S is an assemblage of the disjoint subsets (or blocs) π = {E_{1}, E_{2}, …, E_{l}} such that E_{1} + ⋯ + E_{l} = S. Let bloc π be the number of blocs of π and 𝔓_{n} the set of partitions of [n].

Throughout this paper, we adopt the following vector notations. For n = (n_{1}, …, n_{m}), we define

n! = n_{1}! ⋯ n_{m}!, x^{n} = x_{1}^{n_{1}} ⋯ x^{n_{m}}.

If f(x) := f(x_{1}, …, x_{m}) is a power series in x_{1}, …, x_{m}, we denote by [x^{i}]f(x) the coefficient of x^{i} in f(x). However, for convenience, the coefficient of the monomial x_{1}x_{2} ⋯ x_{m} in f(x) will be denoted by [x]f(x).

Lemma 2. The number of circular permutations on [n] with k maxima is

[z^{k} \frac{x^{n}}{n!} ] log\{cosh xy − y^{−1} sinh xy\}^{−1},

where y = (1 − z)^{1/2}.

Proof. A proof of Lemma 2 can be found in [27, p. 273]. Note that this result is usually attributed to Entringer [16]. In fact, this formula had been given implicitly by André [1] in 1895. See also Dumont [15] for some related results.

Lemma 3. The number of permutations on [n] with k maxima and l cycles is

[γ^{k}η^{l} \frac{x^{n}}{n!} ](cosh xy − y^{−1} sinh xy)^{−η},

where y = \sqrt{1 − γ}.

Proof. This is obvious on combining the exponential formula [18, 30] and Lemma 2.

Theorem 1. We have

(2.12)  ν(x^{n}) = |ℑ_{n}|.
(2.13)  μ(x^{n}) = ∑_{π∈𝔓_{n}} a^{bloc π}.
(2.14)  ψ(x^{n}) = ∑_{σ∈𝔖_{n}}(α + 1)^{cyc σ} = (α + 1)_{n}.
(2.15)  ρ(x^{n}) = (1 − c)^{−n} ∑_{σ∈𝔖_{n}} c^{dec σ}β^{cyc σ}.
(2.16)  φ(x^{n}) = δ^{n} ∑_{σ∈𝔖_{n}}(1 + 1/δ^{2})^{max σ}η^{cyc σ}.

Proof. Formula (2.12) is trivial, since the number of involutions without fixed point on [2n + 1] and [2n] is respectively 0 and 1 · 3 ⋯ (2n − 1). On the other hand, it is well known [13] that the number of permutations of [n] with k cycles is s(n, k), the Stirling number of the first kind, and the number of partitions of [n] with k blocs is S(n, k), the Stirling number of the second kind. Moreover, we have (cf. [13] or [ 36, p. 44])

∑_{k=0}^{∞} ∑_{k=0}^{n} S(n, k)a^{k} \frac{t^{n}}{n!} = exp(a(e^{t} − 1)),

∑_{k=0}^{∞} ∑_{k=0}^{n} s(n, k)β^{k} \frac{t^{n}}{n!} = (1 − t)^{−β}.

The formulae (2.13) and (2.14) follow then by comparing (2.8) and (2.9) with the above identities. In the next place, thanks to Foata’s fundamental transformation [18], we know that ∑_{σ∈𝔖_{n}} c^{dec σ} is equal to the nth Eulerian polynomial [38], of which the exponential generating function is

\frac{1 − c}{1 − ce^{t(1−c)}} .

This and the exponential formula [18, 30] imply that

∑_{n≥0} ( ∑_{σ∈𝔖_{n}} c^{dec σ}β^{cyc σ}) \frac{t^{n}}{n!} = (\frac{1 − c}{1 − ce^{t(1−c)}} )^{β} .

Comparing the above identity with (2.10) yields (2.15). Finally, it follows from Lemma 3 that

∑_{n≥0} δ^{n} ∑_{σ∈S_{n}}(1 + 1/δ^{2})^{max σ}η^{cyc σ} \frac{t^{n}}{n!} = (cosh(it) − \frac{δ}{i} sinh(it))^{−η}
= (cos t − δ sin t)^{−η}.

We obtain (2.16) by comparing the above identity with (2.11).

Remark. Theorem 1 actually gives Viennot’s combinatorial definitions of the moments of Sheffer polynomials [40]. Note that Viennot’s original interpretation of ρ(x^{n}) (respectively φ(x^{n})) uses the number of descents (respectively pics) and saillants; however by applying Foata’s fundamental transformation [18] it is easily seen that these statistics have the same distribution as ( dec, cyc) (respectively ( max, cyc)).

The following results are useful in the new derivations of the interpretation of the l.c. of Sheffer polynomials. Let

(2.17)  f_{m}(γ, η) = ∑_{σ∈𝔇_{n}} γ^{dec σ}η^{cyc σ},
(2.18)  g_{m}(γ, η) = ∑_{σ∈𝔇_{n}} γ^{max σ}η^{cyc σ}.

Proposition 2. We have

(2.19) 1 + ∑_{m≥1} f_{m}(γ, η)\frac{x^{m}}{m!} = \{1 − ∑_{k≥2}(γ + γ^{2} + ⋯ + γ^{k−1})\frac{x^{k}}{k!} \}^{−η},

(2.20) 1 + ∑_{m≥1} g_{m}(γ, η)\frac{x^{m}}{m!} = \{1 − ∑_{k≥2} γ ∑_{l≥0} \binom{k − 1}{2l + 1}(1 − γ)^{l} \frac{x^{k}}{k!} \}^{−η}.

Proof. We just prove (2.20) here since (2.19) can be proved in the same manner, see also [42]. For π ∈ 𝔇_{n}, the weight function w(π) = γ^{maxπ}η^{cycπ} is multiplicative with respect to the cycle decomposition. Note that each derangement π does not contain any fixed point (or cycle of length 1), whence, by the exponential formula (cf. [18]) and Lemma 2 with y = \sqrt{1 − γ}, we have

1 + ∑_{m≥1} d_{m}(γ, η)\frac{x^{m}}{m!} = exp\{η log(cosh xy − y^{−1} sinh xy)^{−1} − ηx\}

= \{\frac{e^{(1+y)x} + e^{(1−y)x}}{2} − y^{−1} \frac{e^{(1+y)x} − e^{(1−y)x}}{2} \}^{−η}

= \{1 − ∑_{k≥2}(1 − y^{2}) ∑_{l≥0} \binom{k − 1}{2l + 1}y^{2l} \frac{x^{k}}{k!} \}^{−η}.

The proof is complete by substituting y^{2} = 1 − γ in the last line.

Corollary 1. We have

∑_{σ∈𝔇_{n}}(δ + i)^{dec σ}(δ − i)^{exc σ}η^{cyc σ} = δ^{n} ∑_{σ∈𝔇_{n}}(1 + 1/δ^{2})^{max σ}η^{cyc σ}.

Proof. It follows from Proposition 2 by noting that dec σ + exc σ = n for σ ∈ 𝔇_{n}.

3. The generating function of the linearization coefficients

In what follows we shall denote by e_{k} the kth (0 ≤ k ≤ m) elementary symmetric polynomial of x_{1}, …, x_{m}, that is,

(3.1)  ∑_{j=0}^{m} e_{j}x^{j} = ∏_{k=1}^{m}(1 + x_{k}x).

By convention, we set e_{k} = 0 if k > m.

Lemma 4. We have

(3.2)  tan (arctan x_{1} + ⋯ + arctan x_{m}) = \frac{∑_{n≥0}(−1)^{n}e_{2n+1}}{∑_{n≥0}(−1)^{n}e_{2n}} .

Proof. This is routine, by induction on m.

Lemma 5. With i = \sqrt{−1}, the following identities hold

(3.3)  (∑_{n≥0}(−1)^{n}e_{2n})^{2} + (∑_{n≥0}(−1)^{n}e_{2n+1})^{2} = ∏_{k=1}^{m}(1 + x_{k}^{2}).

(3.4)  ∑_{n≥0}(−1)^{n}e_{2n} = \frac{1}{2}\{∏_{k=1}^{m}(1 + ix_{k}) + ∏_{k=1}^{m}(1 − ix_{k})\}.

(3.5)  ∑_{n≥0}(−1)^{n}e_{2n+1} = \frac{1}{2i}\{∏_{k=1}^{m}(1 + ix_{k}) − ∏_{k=1}^{m}(1 − ix_{k})\}.

Proof. Substituting x by i and −i in (3.1), we get

(3.6)  ∑_{n≥0}(−1)^{n}e_{2n} + i ∑_{n≥0}(−1)^{n}e_{2n+1} = ∏_{k=1}^{m}(1 + ix_{k}),

(3.7)  ∑_{n≥0}(−1)^{n}e_{2n} − i ∑_{n≥0}(−1)^{n}e_{2n+1} = ∏_{k=1}^{m}(1 − ix_{k}).

Formulae (3.4) and (3.5) are then straightforward. We obtain (3.3) by multiplying (3.6) and (3.7).

Lemma 6. For the linear functional φ of (2.5), the following identity holds : (3.8)

φ(exp x ∑_{k=1}^{m} arctan x_{k}) = ∏_{k=1}^{m}(1 + x_{k}^{2})^{η/2} {∑_{n≥0}(−1)^{n}e_{2n} − δ ∑_{n≥0}(−1)^{n}e_{2n+1}}^{−η} .

Proof. By virtue of (2.15) of Proposition 2, we have

φ(exp x ∑_{k=1}^{m} arctan x_{k}) = ∑_{n≥0} φ(x^{n})\frac{(arctan x_{1} + ⋯ + arctan x_{m})^{n}}{n!}

= {cos(∑_{k=1}^{m} arctan x_{k}) − δ sin(∑_{k=1}^{m} arctan x_{k})}^{−η}

= \{\frac{1 − δ tan(arctan x_{1} + ⋯ + arctan x_{m})}{\sqrt{1 + tan^{2}(arctan x_{1} + ⋯ + arctan x_{m})}}\}^{−η} .

Substituting (3.2) into the last line, we get

φ (exp x ∑_{k=1}^{m} arctan x_{k}) = \{\frac{∑_{n≥0}(−1)^{n}e_{2n} − δ ∑_{n≥0}(−1)^{n}e_{2n+1}}{\sqrt{(∑_{n≥0}(−1)^{n}e_{2n})^{2} + (∑_{n≥0}(−1)^{n}e_{2n+1})^{2}}} \}^{−η} ,

and then (3.8), in view of (3.3).

We are now in a position to compute the generating function of the linearization coefficients of all the Sheffer polynomials and express them in terms of the elementary symmetric functions of x_{1}, …, x_{m} with the help of the generating functions of the moments and the polynomials.

Theorem 2. We have

(3.9)  ∑_{n≥0} H(n)\frac{x^{n}}{n!} = e^{e_{2}}.

(3.10)  ∑_{n≥0} C(n;a)\frac{x^{n}}{n!} = e^{ae_{2}+···+ae_{m}}.

(3.11)  ∑_{n≥0} L(n;α)\frac{x^{n}}{n!} = (1 − e_{2} − 2 e_{3} − ⋯ − (m − 1)e_{m})^{−(α+1)}.

(3.12) ∑_{n≥0} M(n; β,c)\frac{x^{n}}{n!} = (1 − ∑_{k=2}^{m}(c + ⋯ + c^{k−1})e_{k})^{−β}.

(3.13) ∑_{n≥0} P(n; δ,η)\frac{x^{n}}{n!} = \{1 − ∑_{k=2}^{m}(δ^{k} + δ^{k−2}) ∑_{l≥0} \binom{k − 1}{2l + 1}(\frac{−1}{δ^{2}})^{l} e_{k}\}^{−η}.

Proof. Equations (3.9)–(3.12) are already known [6, 41] and also easily to be established using the generating functions of the polynomials themselves. So we give here only the proof of (3.13). According to (1.7) and (1.12), we have

∑_{n≥0} P(n; δ, η)\frac{x^{n}}{n!} = φ(∏_{k=1}^{m} ∑_{n_{k}≥0} P_{n_{k}}(x; δ, η)\frac{x_{k}^{n_{k}}}{n_{k}!} )

= ∏_{k=1}^{m}((1 + δx_{k})^{2} + x_{k}^{2})^{−η/2}φ(exp x ∑_{j=1}^{m} arctan \frac{x_{j}}{1 + δx_{j}} ).

Now we apply Lemma 6 to the above identity and obtain

(3.14) ∑_{n≥0} P(n; δ, η)\frac{x^{n}}{n!} = ∏_{k=1}^{m}(1 + δx_{k})^{−η}\{∑_{n≥0}(−1)^{n}ê_{2n} − δ ∑_{n≥0}(−1)^{n}ê_{2n+1}\}^{−η},

where ê_{k} (k ≥ 0) are the elementary symmetric polynomials of the variables :

\frac{x_{1}}{1 + δx_{1}} , …, \frac{x_{m}}{1 + δx_{m}} .

Next, it follows from (3.3) and (3.4) of Lemma 5 that

(3.15)  ∑_{n≥0}(−1)^{n}ê_{2n} = \frac{1}{2}\{∏_{k=1}^{m} \frac{1 + (δ + i)x_{k}}{1 + δx_{k}} + ∏_{k=1}^{m} \frac{1 + (δ − i)x_{k}}{1 + δx_{k}} \},

(3.16)  ∑_{n≥0}(−1)^{n}ê_{2n+1} = \frac{1}{2i}\{∏_{k=1}^{m} \frac{1 + (δ + i)x_{k}}{1 + δx_{k}} − ∏_{k=1}^{m} \frac{1 + (δ − i)x_{k}}{1 + δx_{k}} \}.

Finally, putting (3.15) and (3.16) into (3.14), we get

∑_{n} P(n; δ, η)\frac{x^{n}}{n!} = \{\frac{i − δ}{2i} ∏_{k=1}^{m}(1 + (δ + i)x_{k}) + \frac{i + δ}{2i} ∏_{k=1}^{m}(1 + (δ − i)x_{k})\}^{−η} (3.17)

= \{1 + \frac{1 + δ^{2}}{2i} ∑_{k=1}^{m}((δ − i)^{k−1} − (δ + i)^{k−1})e_{k}\}^{−η} ,

which implies (3.13) after simplification.

If m = 3, the explicit formulae can be obtained from Theorem 2 for the linearization coefficients of the Sheffer polynomials.

Corollary 2. Let n = (n_{1}, n_{2}, n_{3}), we have the explicit formulae :

(3.18)  H(n) = \{ \begin{matrix}\frac{n_{1}!n_{2}!n_{3}!}{(s−n_{1})!(s−n_{2})!(s−n_{3})!}, & if n_{1} + n_{2} + n_{3} = 2s; \\ 0, & if n_{1} + n_{2} + n_{3} is odd. \end{matrix}
(3.19)  C(n; a) = ∑_{s≥0} \frac{n_{1}!n_{2}!n_{3}!a^{s}}{(s − n_{1})!(s − n_{2})!(s − n_{3})!(n_{1} + n_{2} + n_{3} − 2s)!}·

(3.20)  L(n; α) = ∑_{s≥0} \frac{n_{1}!n_{2}!n_{3}!2^{n_{1}+n_{2}+n_{3}−2s}(β)_{s}}{(s − n_{1})!(s − n_{2})!(s − n_{3})!(n_{1} + n_{2} + n_{3} − 2s)!}·

(3.21) M(n; β, c) = ∑_{s≥0} \frac{n_{1}!n_{2}!n_{3}!(1 + c)^{n_{1}+n_{2}+n_{3}−2s}c^{s}(β)_{s}}{(s − n_{1})!(s − n_{2})!(s − n_{3})!(n_{1} + n_{2} + n_{3} − 2s)!}·

(3.22) P(n; δ, η) = ∑_{s≥0} \frac{n_{1}!n_{2}!n_{3}!(η)_{s}(1 + 1/δ^{2})^{s}δ^{n_{1}+n_{2}+n_{3}}2^{n_{1}+n_{2}+n_{3}−2s}}{(s − n_{1})!(s − n_{2})!(s − n_{3})!(n_{1} + n_{2} + n_{3} − 2s)!} ·

Remark. Formulae (3.18)–(3.21) have appeared in [3, 5], but, to the author’s knowledge, formula (3.22) seems new. Setting n_{3} = 0 in the above corollary, we recover then the orthogonalities of the Hermite, Charlier, Laguerre, Meixner, and Meixner-Pollaczek polynomials.

4. The combinatorics of c-derangements

Let us begin with some multianalogues of the notions and notations introduced in 2. For each i ∈ [m], we define the coloured sets ℕ_{i} = i × ℕ and [n]_{i} = i × [n]. Let a = (i, j) ∈ ℕ_{i}, we call the first term i the colour of a and set c(a) = i. Let ℕ_{∗} = [m] × ℕ denote the disjoint union ℕ_{1} ∪ ⋯ ∪ ℕ_{m}. For n = (n_{1}, …, n_{m}), we note also [n] = [n_{1}]_{1} ∪ ⋯ ∪ [n_{m}]_{m}. Clearly, the set ℕ_{∗} is a total ordered set with respect to the lexicographic order.

It is convenient to identify a permutation π of a set A with the digraph on A having an edge a → π(a) for each a ∈ A. We will refer to an edge beginning at a point of colour i and ending at a point of colour j as an “i → j” edge, calling it pure if i = j and mixed if i ≠ j. A sequence of consecutive edges a → π(a) → π^{2}(a) → ⋯ → π^{k}(a) (k ≥ 0) is called a path of the graph. We say also that this is a c(a)→c(π(a))→c(π^{2}(a))→ ⋯ →c(π^{k}(a)) path when we just take account of the colours.

Let π be a permutation on [n]. An element a ∈ [n] is said to be a c-fixed point of π if c(a) = c(π(a)). A permutation without c-fixed point is said to be a c-derangement on [n]. We denote respectively by 𝔇_{n} and ℑ_{n} the sets of c-derangements and of c-fixed point free involutions on [n]. Similarly, we say that π has a c-excedance (respectively c-decedance) at a ∈ [n] if c(π(a)) > c(a) (respectively c(π(a)) < c(a)) and denote by exc π (respectively dec π) the number of excedances (respectively decedances) of π.

Proposition 3. Let D(n; γ, η) = ∑ γ^{exc π}η^{cyc π} (π ∈ 𝔇_{n}) and n^{∗} be any rearrangement of n = (n_{1}, …, n_{m}), then

(4.1)  D(n; γ, η) = D(n^{∗}; γ, η).

Proof. Without loss of generality we may assume that n^{∗} is a rearrangement of n by just exchanging n_{i} and n_{i+1}, where 1 ≤ i ≤ m − 1. We will prove (4.1) by constructing an explicit bijection Φ : 𝔇_{n} → 𝔇_{n^{∗}} such that exc π = exc Φ(π) and cyc π = cyc Φ(π) for each π ∈ 𝔇_{n}.

Consider any c-derangement π of [n] identified with its digraph G. Recall that any vertex of π is characterized by a pair of natural numbers (i, j), where i (1 ≤ i ≤ m) is the number of the colour to which it belongs and j (1 ≤ j ≤ n_{i}) is its serial number in that colour. Since there is no pure edge in G, each vertex of colour i or i + 1 is necessarily located in either a cycle of which the vertices are only of colour i and i + 1 or one of the following paths :

r_{1}→i→l_{1}, or r_{2}→(i + 1)→l_{2}, (I)

r_{3}→(i→(i + 1))^{k_{1}}→l_{3}, or r_{4}→((i + 1)→i)^{k_{2}}→l_{4}, (II)

r_{5}→(i→(i + 1))^{k_{3}}→i→l_{5}, or r_{6}→((i + 1)→i)^{k_{4}}→(i + 1)→l_{6}, (III)

where the r_{j}, l_{j} ∉ {i, i + 1} for 1 ≤ j ≤ 6 denote the colours of the corresponding vertices and k_{l} ≥ 1 for 1 ≤ l ≤ 4. Apply now the following tranformation Φ to G :

(1) Exchange the serial numbers of each vertex of colour i (resp. i + 1) with those of its image in each path of type (II) and in each cycle consisting only of vertices of colour i and i + 1 ;

(2) Exchange the colours i and i + 1 of the vertices in the paths of types (I)

and (III). Call the new graph G′. It is then readily seen that G′ is a graph of a c-derangement on [n^{∗}]. Denote the corresponding c-derangement by Φ(π). Clearly, the tranformation Φ is a bijection between 𝔇_{n} and 𝔇_{n^{∗}} with the property that cyc π = cyc Φ(π). On the other hand, each excedance of colour i + 1 (resp. i) of π in a path of type (II) or a cycle consisting only of colour i and i + 1 is mapped to an excedance of Φ(π) with the same colour, and the number of excedances contained in each path of type (I) or (II) is invariant with respect to this transformation. Therefore the total number of excedances does not change after this transformation, that is, exc π = exc Φ(π).

Corollary 3. Let E(n; γ, η) = ∑ γ^{dec π}η^{cyc π} (π ∈ 𝔇_{n}) and n^{∗} be any rearrangement of n = (n_{1}, …, n_{m}), then D(n; γ, η) = D(n^{∗}; γ, η).

Proof. It follows from the fact that dec π = n_{1} +⋯+n_{m} − exc π for each π ∈ 𝔇_{n}.

Consider the graph of any c-derangement π of 𝔇_{n}. Take any cycle of π and call X^{0} the hightest ranking vertex of this cycle. Let X^{i} = π(X^{0}) for i ≥ 1. Then this cycle can be linearly presented as

(4.2)  X^{0}→X^{1}→ ⋯ →X^{p−1}(→X^{p} = X^{0}).

Since π is a c-derangement, it follows that p ≥ 2 and c(X^{i}) ≠ c(X^{i+1}) for 0 ≤ i ≤ p − 1.

A linear arrangement w = a_{1}a_{2} … a_{n} of the elements of a subset A ⊆ ℕ_{∗} is said to be a wave arrangement of A, if no two adjacent elements of w have the same colour, i.e., c(a_{i}) ≠ c(a_{i+1}) for 1 ≤ i ≤ n − 1. A wave arrangement w = a_{1}a_{2} … a_{n} is called a wave segment if moreover n ≥ 2 and c(a_{1}) > c(a_{i}) for all 2 ≤ i ≤ n. In what follows, we will identify w = a_{1}a_{2} … a_{n} with the sequence a_{1}→a_{2}→ ⋯ →a_{n} and call it a c(a_{1})→c(a_{2})→ ⋯ →c(a_{n}) arrangement or segment. It follows that the cycle (4.2) can be divided up into wave segments each of which begins with vertices of the same colour as X^{0}. For example, if the cycle (4.2) is

(7, 6)→(2, 5)→(3, 7)→(7, 1)→(2, 4)→(1, 1)→(4, 3)→(7, 2)→(2, 1).

We should have the segments

w_{1} = (7, 6)→(2, 5)→(3, 7),
w_{2} = (7, 1)→(2, 4)→(1, 1)→(4, 3),
w_{3} = (7, 2)→(2, 1).

Thus, each cycle of π corresponds in this way to a set of wave segments and a circular permutation acting on it. Let C_{i} be the set of cycles of π of which the highest ranking vertices are of colour i (2 ≤ i ≤ m). By the above decomposition of cycles, each class C_{i} (2 ≤ i ≤ m) corresponds to a pair (Ω_{i}, σ_{i}) where Ω_{i} is a set of wave segments beginning with a vertex of colour i and σ_{i} a permutation acting on it. Taking these in count, any c-derangement π can be identified with a (m − 1)-tuple ((Ω_{2}, σ_{2}), …, (Ω_{m}, σ_{m})), with the product (juxtaposition) of all the wave segments in Ω_{2} ∪ … ∪ Ω_{m} being a rearrangement of [n] and the permutations σ_{i} (2 ≤ i ≤ m) acting on Ω_{i}.

To determine the transfer of the weight through this correspondence, we are led to introduce the corrsponding weight on the set of wave segments. Let w = a_{1}a_{2} … a_{n} be a wave segment. By abuse of language, we call a_{i} (1 ≤ i ≤ n) c-decedance if c(a_{i−1} < c(a_{i}) > c(a_{i+1}) with the convention that a_{0} = a_{n} and a_{n+1} = a_{1}. Denote by dec w the number of c-decedances of w and for a set of wave arrangements Ω, define dec Ω = ∑_{w∈Ω} dec w.

Proposition 4. The wave segment decomposition, that is,

π ⟼ ((Ω_{2}, σ_{2}), …, (Ω_{m}, σ_{m})) ,

has the following properties :

dec π = ∑_{i=2}^{m} dec Ω_{i}  and  cyc π = ∑_{i=2}^{m} cyc σ_{i}.

Proof. Firstly, it is readily seen that cyc π = ∑_{i=2}^{m} cyc w_{i}. It then suffices to consider the case where π is a circular c-derangement. Let π ∈ 𝔇_{n} be a circular derangement with the unique circle x^{0}→x^{1}→ ⋯ →x^{p}→x^{0}, where x^{0} is the highest ranking vertex and x^{l} = π^{l}(x^{0}) for 1 ≤ l ≤ p. Let Ω = {w_{1}, w_{2}, …, w_{k}} be the wave segment decomposition of π with

w_{1} = a_{0}a_{1} ⋯ a_{l_{1}−1},
w_{2} = a_{l_{1}}a_{l_{1}+1} ⋯ a_{l_{2}−1},
· · · · · ·
w_{k} = a_{l_{k}}a_{l_{k}+1} ⋯ a_{p},

where a_{l} = x^{l} for 0 ≤ l ≤ p and c(x^{0}) = c(x^{l_{1}}) = … = c(x^{l_{k}}). Then the associated circular permutation σ acting on Ω is w_{1}→w_{2}→ ⋯ →w_{k}→w_{1}. Clearly, an element x^{l} is a c-decedance of π if and only if a_{l} is a c-decedance of some w_{i}. Hence dec π = ∑_{i=1}^{k} dec w_{i}.

5. The generating functions of c-derangements

Theorem 3. The following identities hold :

(5.1)  ∑_{n≥0} |ℑ_{n}| \frac{x^{n}}{n!} = e^{e_{2}};

(5.2)  ∑_{n≥0} ( ∑_{π∈𝔓_{n}} a^{bloc π}) \frac{x^{n}}{n!} = e^{ae_{2}+···+ae_{m}};

(5.3)  ∑_{n≥0} ( ∑_{π∈𝔇_{n}} β^{cyc π}) \frac{x^{n}}{n!} = (1 − ∑_{k=2}^{m}(k − 1)e_{k})^{−β} ;

(5.4)  ∑_{n≥0} ( ∑_{π∈𝔇_{n}} c^{dec π}β^{cyc π}) \frac{x^{n}}{n!} = (1 − ∑_{k=2}^{m}(c + ⋯ + c^{k−1})e_{k})^{−β} ;

(5.5) ∑_{n≥0} ( ∑_{π∈𝔇_{n}}(δ − i)^{exc π}(δ + i)^{dec π}η^{cyc π}) \frac{x^{n}}{n!}

= \{1 − ∑_{k=2}^{m}(δ^{k} + δ^{k−2}) ∑_{l≥0} \binom{k − 1}{2l + 1} (\frac{−1}{δ^{2}})^{l} e_{k}\}^{−η} .

Proof. Equations (5.1) − (5.4) are already known [20, 28, 40]. In fact, formulae (5.1) and (5.2) are straightforward consequences of the multivariable exponential formula [30], and formulae (5.3) and (5.4) were proved in [20, 41] by applying the βextension of MacMahon’s Master Theorem. Besides, if c = 1, formula (5.4) reduces to (5.3). So in the sequel only the proof of (5.5) will be given. First we note that for each π ∈ 𝔇_{n}, we have

exc π + dec π = n_{1} + ⋯ + n_{m}.

Hence, in (5.4) if we substitute c by (δ + i)/(δ − i), β by η and x_{k} by (δ − i)^{k}, we get immediately

∑_{n≥0} ( ∑_{π∈𝔇_{n}}(δ − i)^{exc π}(δ + i)^{dec π}η^{cyc π}) \frac{x^{n}}{n!}

= \{1 − \frac{δ^{2} + 1}{2i} ∑_{k=2}^{m} [(δ + i)^{k−1} − (δ − i)^{k−1}] e_{k}\}^{−η} ,

which yields (5.5) after simplification.

In view of the importance of (5.4), we give here two more proofs of this formula.

Lemma 7. Let Φ(x_{1}, x_{2}, …) be a power series in x_{1}, x_{2}, …, then

[x]Φ(e_{1}, e_{2}, …) = [\frac{x^{m}}{m!} ] Φ(\frac{x^{1}}{1!} , \frac{x^{2}}{2!} , …,).

For a proof of this result, we refer the reader to [27, p. 233].

First New Proof of (5.4). We will count the c-derangements through the wave segment decomposition given in Proposition 4. For 1 ≤ i ≤ m − 1, let n_{i} = (n_{1}, …, n_{i}) and W_{i} be the set of wave arrangements of [n_{1}] ∪ … ∪ [n_{i}]. Let π = a_{1} … a_{n} be a wave arrangement of W_{i} and α ∈ ℕ_{∗} be an element with colour c(α) = i + 1, it is then clear that απ = α a_{1} … a_{n} is a wave segment beginning with colour i + 1. Define the generating function for W_{i} by

(5.6)  f_{i}(x_{1}, …, x_{i}; γ) = ∑_{n_{i}} ( ∑_{π∈W_{i}} γ^{dec π}) \frac{x_{1}^{n_{1}}}{n_{i}!} ⋯ \frac{x_{m}^{n_{m}}}{n_{m}!} .

It follows that the generating function of the wave segments beginning with colour i+1 should be γ x_{i+1}f_{i}(x_{1}, …, x_{i}; γ). On the other hand, since each c-derangement π corresponds bijectively to a (m−1)-tuple ((Ω_{2}, σ_{2}), …, (Ω_{m}, σ_{m})) and the generating function of permutations with each cycle being weighted by η is (1−x)^{−η}, the generating function of (Ω_{i+1}, σ_{i+1}) should be (1 − γ x_{i+1}f_{i}(x_{1}, …, x_{i}; γ))^{−η} where 1 ≤ i ≤ m − 1. Thanks to Proposition 4, we obtain

(5.7)  ∑_{n≥0} D(n; γ, η)\frac{x^{n}}{n!} = ∏_{i=1}^{m−1}(1 − γ x_{i+1}f_{i}(x_{1}, …, x_{i}; γ))^{−η}.

Moreover, by Corollary 3, both sides of (5.7) are symmetric functions of x_{1}, …, x_{m}, which implies that the function

(5.8)  ∏_{i=1}^{m−1}(1 − γ x_{i+1}f_{i}(x_{1}, …, x_{i}; γ))

is also symmetric. Furthermore, we note that (5.8) is actually linear in x_{m}. By symmetry, it is a linear symmetric function of x_{1}, …, x_{m}. We can then expand (5.8) as a linear combination of the elementary symmetric functions e_{i} (0 ≤ i ≤ m) (cf. [33])

(5.9)  ∏_{i=1}^{m−1}(1 − γ x_{i+1}f_{i}(x_{1}, …, x_{i}; γ)) = a_{0} + a_{1}e_{1} + ⋯ + a_{m}e_{m},

where the coefficients a_{i} (0 ≤ i ≤ m) are to be determined. Using (5.9) we can rewrite (5.7) as

(5.10)  ∑_{n≥0} D(n; γ, η)\frac{x^{n}}{n!} = (a_{0} + a_{1}e_{1} + ⋯ + a_{m}e_{m})^{−η}.

Applying Lemma 7 to (5.10) and noticing that the coefficient of x_{1} … x_{m} in the left-hand side of (5.10) is the generating polynomial d_{m}(γ, η) of the ordinary derangements of 𝔇_{m} (cf. Proposition 2), we obtain the following equation :

\{1 − ∑_{k≥2}(γ + γ^{2} + ⋯ + γ^{k−1})\frac{x^{k}}{k!} \}^{−η} = (∑_{k≥0} a_{k} \frac{x^{k}}{k!} )^{−η} .

Equating the coefficients of x^{n} gives a_{0} = 1, a_{1} = 0 and for k ≥ 2

a_{k} = −(γ + γ^{2} + ⋯ + γ^{k−1}).

Finally, substituting these values into (5.10) yields (5.4).

Second New Proof of (5.4). Let f_{i}(γ) = f_{i}(x_{1}, …, x_{i}; γ), it is then well-known [27, p.78] that

f_{i}(γ) = \frac{∏_{j=1}^{i}(1 + γ x_{j}) − ∏_{j=1}^{i}(1 + x_{j})}{γ ∏_{j=1}^{i}(1 + x_{j}) − ∏_{j=1}^{i}(1 + γ x_{j})}

where 1 ≤ i ≤ m − 1. It follows that

(5.11)  (1 − γ x_{i+1}f_{i}(γ))^{−1} = \frac{γ ∏_{j=1}^{i}(1 + x_{j}) − ∏_{j=1}^{i}(1 + γ x_{j})}{γ ∏_{j=1}^{i+1}(1 + x_{j}) − ∏_{j=1}^{i+1}(1 + γ x_{j})}.

Substituting this in (5.7) we get

∑_{n≥0} D(n; γ, η)\frac{x^{n}}{n!} = ∏_{i=1}^{m−1}(1 − γ x_{i+1}f_{i}(γ))^{−β}

= ∏_{i=1}^{m−1} (\frac{γ ∏_{j=1}^{i}(1 + x_{j}) − ∏_{j=1}^{i}(1 + γ x_{j})}{γ ∏_{j=1}^{i+1}(1 + x_{j}) − ∏_{j=1}^{i+1}(1 + γ x_{j})})^{−β}

= \{\frac{γ(1 + x_{1}) − (1 + γ x_{1})}{γ ∏_{j=1}^{m}(1 + x_{j}) − ∏_{j=1}^{m}(1 + γ x_{j})}\}^{−β}

= (1 − γ e_{2} − (γ + γ^{2}) e_{3} − … − (γ + ⋯ + γ^{m−1}) e_{m})^{−β},

as desired.

6. Combinatorial interpretations of the linearization coefficients

Combining Theorems 2 and 3, we immediately derive the following combinatorial interpretations of the linearization coefficients of the orthogonal Sheffer polynomials.

Thoerem 4. The following identities hold :

(6.1)  H(n) = |ℑ_{n}|;
(6.2)  C(n; a) = ∑_{π∈𝔓_{n}} a^{bloc π};
(6.3)  L(n; α) = ∑_{π∈𝔇_{n}}(1 + α)^{cyc π};
(6.4)  M(n; β, c) = ∑_{π∈𝔇_{n}} c^{dec π}β^{cyc π};
(6.5)  P(n; δ, η) = ∑_{π∈𝔇_{n}}(δ + i)^{dec π}(δ − i)^{exc π}η^{cyc π};

As mentioned in Section 1, formula (6.1)–(6.4) have been established by other methods in [8, 14, 20, 41]. One notices that Theorem 4 shows that the linearization coefficients of Hermite, Charlier, Laguerre and Meixner polynomials are actually the polynomials of some appropriate variables with nonnegative integral coefficients. Therefore, the classical positivities (cf. [4]) of these coefficients become obvious. However, although it is evident from (6.5) that P(n; δ, η) is a polynomial of δ and η, the fact that the coefficients are nonnegative integers is not obvious.

Proposition 5. The coefficients of the polynomial P(n; δ, η) of δ and η are nonnegative integers.

proof. We first prove that it is a polynomial with integral coefficients. Cearly, it is sufficient to prove that in the summation of (6.5) each term appears with its conjugate. For this end, we construct an involution T on 𝔇_{n} as follows. For each π ∈ 𝔇_{n}, we define T (π) = π′ by π′(a) = b if and only if π(b) = a, where a, b are elements of [n]. It is easy to see that T is an involution without fixed point on 𝔇_{n} such that

(6.6)  cyc π = cyc π′, dec π = exc π′, exc π = dec π′.

Therefore, in the summation of (6.5), the summand corresponding to π′ is the conjugate of that corresponding to π′. It follows that P(n; δ, η) is a polynomial in δ and η with integral coefficients. It remains to prove that the coefficients are positive. We shall prove this for m ≤ 4, for the general cases are similar to the case m = 4. Set P_{n} := P_{n}(x; δ, η). By (3.22) we have

(6.7)  φ(P_{n_{1}}P_{n_{2}}P_{n_{3}}) = ∑_{s≥0} \frac{n_{1}!n_{2}!n_{3}!(η)_{s}(δ^{2} + 1)^{s}(2 δ)^{n_{1}+n_{2}+n_{3}−2s}}{(s − n_{1})!(s − n_{2})!(s − n_{3})!(n_{1} + n_{2} + n_{3} − 2s)!}·

Therefore, the proposition is true for m ≤ 3. For m = 4 we derive from the linearization formula (1.0) that

(6.8)  φ(P_{n_{1}}P_{n_{2}}P_{n_{3}}P_{n_{4}}) = ∑_{k≥0} \frac{φ(P_{k}P_{n_{1}}P_{n_{2}})φ(P_{k}P_{n_{3}}P_{n_{4}})}{φ(P_{k}P_{k})}·

By (6.7) and noting that φ(P_{k}P_{k}) = (k!)^{2}(η)_{k}(δ^{2} + 1)^{k}, we may conclude that (6.8) is a polynomial with positive coefficients.

When δ = 0, the generating function (1.7) reduces to

(6.9)  ∑_{n≥0} M_{n}(x; 0, η)\frac{t^{n}}{n!} = (1 + t^{2})^{−η/2} exp(x arctan t).

Note that the polynomials i^{n} M_{n}(−ix; 0, η) were also introduced by Carlitz [11]. We first derive some consequences from Theorem 4 for P(n; 0, η) before we proceede to give another interesting interpretation in a different context.

Proposition 6. If n = n_{1} + ⋯ + n_{m} is odd, then P(n; 0, η) = 0.

Proof. Recall that the involution T defined in the proof of Proposition 5 has the following properties : if T (π) = π′, then

(6.10)  cyc π = cyc π′, exc π = n − exc π′.

Whence, if n = n_{1} + ⋯ + n_{m} is odd, in the sum of (6.5) the weight of π is killed by that of π′, i.e.,

(−1)^{exc π}i^{n}η^{cyc π} + (−1)^{exc π′}i^{n}η^{cyc π′} = 0.

Thus the total sum (6.5) is clearly zero.

Let 𝔈_{n} be the set of permutations of 𝔇_{n} of which all the cycles are of even length. It follows that 𝔈_{n} = ∅ if n = n_{1} + ⋯ + n_{m} is odd.

Proposition 7. We have

(6.11)  P(n; 0, η) = ∑_{π∈𝔈_{n}}(−1)^{exc π}i^{n}η^{cyc π}.

Proof. Clearly, if we can define an involution θ : 𝔇_{n} \ 𝔈_{n} ⟶ 𝔇_{n} \ 𝔈_{n} such that θ(π) = π′ with

(6.12)  cyc π = cyc π′ and exc π + exc π′ is odd.

The identity (6.10) would follow immediately. For each π ∈ 𝔇_{n} \ 𝔈_{n}, we may assume that the set of cycles of π is totaly ordered (for example, according to the order of the highest ranking element of each cycle). Among the cycles of π, let a_{1} → a_{2} → ⋯ → a_{r} be the least cycle with odd length r, then π′ = θ(π) is defined by the same cycles as π, except that the least cycle of odd length is a_{r} → a_{r−1} → ⋯ → a_{1}. Clearly θ is an involution on 𝔇_{n} \ 𝔈_{n} and satisfies (6.12), for exc π + exc π′ ≡ r ≡ 1 (mod 2).

The new interpretation of P(n; 0, η) relies on the model of generalized up-down permutations. Hereafter, we consider a permutation of any finite subset A of ℕ_{∗} as a linear arrangement a_{1}a_{2} ⋯ a_{n} of its elements. A permutation π = a_{1}a_{2} ⋯ a_{n} is said to be up-down if c(a_{1}) < c(a_{2}) ≥ c(a_{3}) < ⋯ _{≥}^{<}c(a_{n}), and basic if it begins with its smallest element. Also, we say that the length of the permutation π = a_{1}a_{2} ⋯ a_{n} is n. Let 𝔖_{n} be the set of all permutations on [n] and 𝔘_{n} the set of up-down permutations on [n]. A record of a permutation π = a_{1}a_{2} ⋯ a_{n} is an a_{j} such that i < j implies a_{i} > a_{j}. Denote by rec π the number of records of π and by F (π) the first element of π. A permutation π = a_{1}a_{2} ⋯ a_{n} is basic if it begins with its smallest element. Let 𝔅_{n} be the set of basic permutations of 𝔖_{n}.

As noted by Gessel [24], every permutation π = a_{1} a_{2} ⋯ a_{n} has a unique factorization π = β_{1}β_{2} ⋯ β_{k} such that each β_{i} is basic and F (β_{1}) > F (β_{2}) > ⋯ > F (β_{k}). Such a factorization is called basic decomposition of π and the β_{i} the basic components of π. We call permutation reduced if it is a permutation of [n]. To any permutation π = a_{1}a_{2} ⋯ a_{n} we may associate a reduced permutation, red(π), by replacing in π the jth smallest element of color i by (i, j). Thus

red((1, 7)(2, 9)(1, 2)(2, 6)) = (1, 2)(2, 2)(1, 1)(2, 1).

The content of the permutation π = a_{1}a_{2} ⋯ a_{n} is the set con(π) = {a_{1}, a_{2}, …, a_{n}}. We note that a permutation is determined by its reduction and its content.

A function ω defined on permutations (with values in some commutative algebra over the rationals) is multiplicative if for all permutations π :

(1) ω(π) = ω(red(π)).

(2) If β_{1}β_{2} ⋯ β_{k} is the basic decomposition of π, then

ω(π) = ω(β_{1}) ω(β_{2}) ⋯ ω(β_{k}).

Thus a multiplicative function is determined by its values on reduced basic permutations, and these may be chosen arbitrarily. One of the fundamental properties of a multiplicative function is the so-called exponential formula. The proof is similar to the case of one variable.

Lemma 8. Let ω be a multiplicative function on permutations. For any sequence n ≥ 0, set g_{n} = ∑_{π∈𝔖_{n}} ω(π) and f_{n} = ∑_{π∈𝔅_{n}} ω(π), then

∑_{n≥0} g_{n} \frac{x^{n}}{n!} = exp ∑_{n>0} f_{n} \frac{x^{n}}{n!} .

To each permutation π of [n] we associate the weight w(π) = η^{rec(π)} if π is updown and of even length or 0 otherwise. Clearly S_{n}(η) = ∑_{π∈§_{n}} w(π) and B_{n}(η) = ∑_{π∈𝔅_{n}} w(π) count respectively the up-down and basic up-down permutations of even length according to the records.

Lemma 9. We have

(6.13)  ∑_{n≥0} S_{n}(η)\frac{x^{n}}{n!} = (∑_{n≥0}(−1)^{n}e_{2n})^{−η}.

Proof. As the weight w(π) is multiplicative, by Lemma 8, we have

(6.14)  ∑_{n≥0} S_{n}(η)\frac{x^{n}}{n!} = exp ∑_{n>0} B_{n}(η)\frac{x^{n}}{n!} = exp η ∑_{n>0} B_{n}(1)\frac{x^{n}}{n!} .

But it is well-known [Go-Ja, p. 234] that

exp ∑_{n>0} B_{n}(1)\frac{x^{n}}{n} = ∑_{n≥0} S_{n}(1)\frac{x^{n}}{n!} = (∑_{n≥0}(−1)^{n}e_{2n})^{−1}.

The proof is then complete by putting the last identity into (6.14).

Theorem 5. For any n = (n_{1}, …, n_{m}), the polynomials P(n; 0, η) are the generating polynomials of up-down permutations of even length on [n] weighted by records, viz.,

P(n; 0, η) = \{ \begin{matrix}∑_{π∈𝔘_{n}} η^{rec π}, & if n_{1} + ⋯ + n_{m} is even, \\ 0, & if n_{1} + ⋯ + n_{m} is odd. \end{matrix}

Proof. If δ = 0, the right-hand side of (3.13) reduces to that of (6.13). The Theorem follows then from Lemma 9 and Theorem 2.

By comparison of Proposition 6 with Theorem 5, we derive the following

Corollary 4. We have

(6.15)  ∑_{π∈𝔈_{n}}(−1)^{exc π}i^{n}η^{cyc π} = ∑_{π∈𝔘_{n}} η^{rec π}.

Remark. When n = (1, 1, …, 1) and η = 1, identity (6.15) reduces to a well-known result [43, Th. 4.7].

In the case where η = 0, the polynomials p_{n}(x) := p_{n}(x; 0, 0) are no longer orthogonal with respect to φ. However, there is another linear functional ζ of some interest, which may be defined by the generating function of the sequence ζ(x^{n}) (n ≥ 0) :

(6.16)  ∑_{n≥0} ζ(x^{n})\frac{x^{n}}{n!} = tan x.

It then follows from Lemma 5 that

ζ(exp x ∑_{k=1}^{m} arctan x_{k}) = \frac{∑_{n≥0}(−1)^{n}e_{2n+1}}{∑_{n≥0}(−1)^{n}e_{2n}} .

In other words, the following identity holds :

(6.17)  ∑_{n} ζ(∏_{k=1}^{m} p_{n_{k}}(x))\frac{x^{n}}{n!} = \frac{∑_{n≥0}(−1)^{n}e_{2n+1}}{∑_{n≥0}(−1)^{n}e_{2n}} .

However, it is well-known that the right hand side of (6.17) is the generating function of the numbers of up-down permutations of odd length on [n] (cf. [27, p. 196]). So, ζ(∏_{k=1}^{m} p_{n}(x)) counts the up-down permutations of odd length on [n]. Especially, if m = 2, we obtain the quasi-orthogonality :

ζ(p_{n_{1}}(x)p_{n_{2}}(x)) = \{ \begin{matrix}n_{1}! n_{2}!, & if |n_{1} − n_{2}| = 1; \\ 0, & otherwise. \end{matrix}

These results may be of some interest in the theory of coefficients extraction of symmetric functions (cf. [23]).

Acknowledgements

This work was done while the author was visiting Institute for Advanced Study as a member in 1989-1990 and supported by NSF Grant DMS-8610730. The author is also indebted to Professor Jet Wimp for helpful discussions and correspondence regarding Meixner-Pollaczek polynomials, which led particularly to the discovery of Lemma 1.

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