Séminaire Lotharingien de Combinatoire 76 (2017), Article B76b

LACUNARY GENERATING FUNCTIONS FOR THE LAGUERRE POLYNOMIALS

D. BABUSCI, G. DATTOLI, K. GÓRSKA, AND K. A. PENSON

Abstract. Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Laguerre polynomials for which we give a number of new closed form expressions. We present furthermore the different possibilities offered by the method we have developed, with particular emphasis on their link to a new family of special functions and with previous formulations, associated with the theory of quasi monomials.

1. Introduction

1.1. Motivation. This work deals with derivation of a number of summation formulas involving ordinary and generalized Laguerre polynomials of degree n, denoted by L_{n}(x) and L_{n}^{(α)}(x), respectively. More specifically, we are interested in ordinary, exponential and more general generating functions for polynomials L_{2n}(x), L_{3n}(x), L_{2n+l}(x), L_{3n+l}(x) etc., and for their generalized counterparts. In this paper they will be called lacunary generating functions. The Laguerre polynomials appear quite naturally in the theory of the following differential operator

D_{x}^{(α)} = \frac{d}{dx}x\frac{d}{dx} + α\frac{d}{dx},  (1.1)

where x is the indeterminate and α is a real parameter [7, 12, 13, 22]. The specific context is a less-known [21] formulation of theory of polynomials P_{n}(x, λ) (of degree n in x) which is based on the action of the exponential of a certain differential operator Ô(x, \frac{d}{dx}) on monomials x^{n} through

exp[λ Ô(x, \frac{d}{dx})]x^{n} = P_{n}(x, λ), (n = 0, 1, …).  (1.2)

Eq. (1.2) is the defining relation of a family of two-variable polynomials and P_{n}(x, λ) appears to be related to umbral calculus [29, 20]. For the two-variable Hermite polynomials H_{n}^{(2)}(x, λ) the following relation holds [7, 13]:

exp[λ \frac{d^{2}}{dx^{2}}]x^{n} = H_{n}^{(2)}(x, λ),  (1.3)

where the polynomials H_{n}^{(2)}(x, λ) are related to the conventional Hermite polynomials H_{n}(x) through

H_{n}^{(2)}(x, λ) = (−i \sqrt{λ})^{n}H_{n}(\frac{i x}{2 \sqrt{λ}}),  H_{n}(x) = 2^{n}H_{n}^{(2)}(x, −1/4).  (1.4)

Relation (1.2) for the operator Ô(x, \frac{d}{dx}) = D_{x}^{(α)} reads (see, e.g., [22])

exp[λ D_{x}^{(α)}]x^{n} = n! L_{n}^{(α)}(λ, x) = n! λ^{n}L_{n}^{(α)}(−x/λ),  (1.5)

which justifies the name ‘Laguerre derivative’ for D_{x}^{(α)} ^{1}. Eq. (1.5) was obtained using the fact, easily shown by induction,

[D_{x}^{(α)}]^{k}x^{n} = \frac{n!}{(n − k)!} (1 + α − n − k)_{k}x^{n−k},  (1.6)

with (a)_{n} = Γ(a + n)/Γ(a) being the Pochhammer symbol. It generalizes Eq. (22) of [22] to α 0. As an immediate consequence of Eq. (1.6) the following relation obtains:

exp[λ D_{x}^{(α)}]e^{−x} = \frac{1}{(1 + λ)^{1+α}} exp(\frac{−x}{1 + λ} ),  (1.7)

which is equivalent to one of the standard generating functions for the Laguerre polynomials.

In general, the differential operator Ô(x, \frac{d}{dx}) can be seen as part of the evolution equation for a function f (x, t), given by a partial differential equation of the following type:

\frac{∂}{∂t}f (x, t) = κ Ô(x, \frac{∂}{∂x}) f (x, t), x > 0,  (1.8)

where κ is a coupling constant (assuming here κ = 1), which for the operator Ô(x, \frac{d}{dx}) ≡ −D_{x}^{(α)} becomes the Cauchy problem with the initial condition f (x) [9]:

\frac{∂}{∂t}f_{α}(x, t) = − (\frac{∂}{∂x}x\frac{∂}{∂x} + α\frac{∂}{∂x}) f_{α}(x, t),  (1.9)
f_{α}(x, 0) = f (x).  (1.10)

The formal solution of Eqs. (1.8) and (1.10) is obtained via

f_{α}(x, t) = exp[−t D_{x}^{(α)}]f (x).

Therefore Eq. (1.7) for λ ≡ t describes the exact time evolution under the Laguerre derivative D_{x}^{(α)} from the initial condition f (x) = e^{−x}. Whereas Eq. (1.7) could have been obtained via conventional and well-known formula for ∑ _{n=0}^{∞} x^{n}L_{n}^{(α)}(x), this is not anymore the case for initial conditions differing from the exponential.

We illustrate this situation by choosing, e.g., f (x) = x^{−α/2} I_{α/2}(β x), with α, β > 0, respectively f (x) = I_{0}(β x), where I_{α/2}(z) is the modified Bessel function of the first kind,

I_{α/2}(β x) = ∑_{n=0}^{∞} \frac{(β x/2)^{2n+α/2}}{n! Γ(1 + α/2 + n)} .

We obtain

exp[−t D_{x}^{(α)}]x^{−α/2}I_{α/2}(β x) = \frac{(β/2)^{α/2}}{Γ(1 + α/2)} ∑_{n=0}^{∞} \frac{(1/2)_{n}}{(1 + α/2)_{n}} (β t)^{2n}L_{2n}^{(α)}(x/t),  (1.11)

exp[−t D_{x}^{(α)}] I_{0}(β x) = ∑_{n=0}^{∞} \frac{(1/2)_{n}}{n!} (β t)^{2n}L_{2n}^{(α)}(x/t).  (1.12)

Any further evaluation of Eqs. (1.11) and (1.12) turned out to be impossible. In addition, an extensive search in the literature for the appropriate formula gave no results.

We have considered it therefore as our objective to establish the summation formulas of type Eqs. (1.11) and (1.12), and more general ones, which involved lacunary Laguerre series.

1Strictly speaking, Laguerre polynomials are one-variable polynomials and are defined as L_{n}(x) = L_{n}(1, −x).

It is also easily checked that L_{n}(x, y) = x^{n}L_{n}(−\frac{y}{x}). We will use two-variable forms for future convenience. In final results we shall again mostly use the one-variable form.

(Some lacunary exponential generating functions for other polynomials are known: compare [27] for H_{2n}(x), [15] for H_{3n}(x), and various lacunary generating functions for Legendre and Chebyshev polynomials [17]).

Note that the differential equations of type (1.8) were already considered in a combinatorial context as a tool to derive generating functions for combinatorial graphs associated with Ô(x, \frac{d}{dx}), see [22].

1.2. Description of the method. It has been shown that symbolic methods of umbral nature provide powerful tools to deal with the properties of special polynomials and functions [3, 4, 5, 8]. These techniques greatly simplify the problems underlying such studies and allow to reduce the derivation of the relevant properties to straightforward algebraic manipulations.

Similarly to classical references on the umbral calculus [29, 20], we start our considerations with the shift operator c_{z} defined here by

c_{z}^{α} : f (z) ↦ f (z + α),  (1.13)

satisfying c_{z}^{α}c_{z}^{β} = c_{z}^{α+β}. When c_{z}^{α} acts on [Γ(1 + z)]^{−1}, the result evaluated at z = 0 gives [Γ(1 + α)]^{−1}. In this context the ordinary Laguerre polynomials L_{n}^{(0)}(x) ≡ L_{n}(x) can be obtained with the help of c_{z}^{α} as

L_{n}(x)= (1 − xc_{z})^{n} \frac{1}{Γ(1 + z)} |_{z=0}  (1.14)

= ∑_{k=0}^{n} \binom{n}{k}(−x)^{k}c_{z}^{k} \frac{1}{Γ(1 + z)} |_{z=0}

= ∑_{k=0}^{n} \binom{n}{k}(−x)^{k} \frac{1}{Γ(1 + k + z)} |_{z=0}

= ∑_{k=0}^{n} \binom{n}{k} \frac{(−x)^{k}}{k!}.

In the same vein, the two-variable generalized Laguerre polynomials

L_{n}^{(α)}(x, y)= \frac{1}{n!} ∑_{k=0}^{n} \binom{n}{k}(1 + α + k)_{n−k} x^{n−k} y^{k}

= \frac{Γ(1 + α + n)}{n!} Λ_{n}^{(α)}(x, y)  (1.15)

are obtained via

Λ_{n}^{(α)}(x, y) = ∑_{k=0}^{n} \binom{n}{k} \frac{x^{n−k} y^{k}}{Γ(1 + α + k)}  (1.16)

= ∑_{k=0}^{n} \binom{n}{k}x^{n−k} y^{k}c_{z}^{k} \frac{1}{Γ(1 + z)} |_{z=α}

= (x + y c_{z})^{n} \frac{1}{Γ(1 + z)} |_{z=α}  (1.17)
= c_{z}^{α} (x + y c_{z})^{n} \frac{1}{Γ(1 + z)} |_{z=0}.  (1.18)

Note that with our definition of L_{n}^{(α)}(x) given in Eqs. (1.15) with (1.17) or (1.18) we can do about the same as with the definition à la Rota (see p. 198 in [20])^{2}, namely L_{n}^{(α)}(x) = x^{−α} (\frac{d}{dx} − I)^{n}x^{n+α}.

Using the method exposed in Eqs. (1.14) and (1.15), the Laguerre polynomials L_{2n}(x, y) and L_{3n}(x, y) can be expressed via Λ_{n}^{(α)}(x, y) as follows:

L_{2n}(x, y) = (x + y c_{z})^{2n} \frac{1}{Γ(1 + z)} |_{z=0} = ∑_{r=0}^{n} \binom{n}{r}x^{n−r} y^{r} Λ_{n}^{(r)}(x, y),  (1.19)

and

L_{3n}(x, y) = (x + y c_{z})^{3n} \frac{1}{Γ(1 + z)} |_{z=0} = ∑_{k=0}^{n} ∑_{r=0}^{n} \binom{n}{r} \binom{n}{k}x^{2n−r−k} y^{r+k} Λ_{n}^{(r+k)}(x, y).

A further family of polynomials, introduced by the same means as before, is provided by what we will call, for reasons which will become clear in the following, the two-parameter family of Laguerre–Wright polynomials, namely

Λ_{n}^{(α,β)}(x, y) = (x + y c_{z}^{β})^{n} \frac{1}{Γ(1 + z)} |_{z=α} = ∑_{r=0}^{n} \binom{n}{r} \frac{x^{n−r} y^{r}}{Γ(1 + α + β r)} .  (1.20)

From Eq. (1.17) Λ_{n}^{(α,1)}(x, y) = Λ_{n}^{(α)}(x, y) follows.

2. Standard generating functions of Laguerre polynomials

The first family of examples will concern standard generating functions of polynomials Λ^{(α)}(x) and Λ^{(α,β)}(x). In this way we illustrate the usefulness of our approach. We will consider in the following two types of generating functions, namely the exponential and the ordinary ones. In the first case, Eq. (1.20) leads to the following form

∑_{n=0}^{∞} \frac{t^{n}}{n!} Λ_{n}^{(α,β)}(x, y) = ∑_{n=0}^{∞} \frac{1}{n!}[t (x + y c_{z}^{β})]^{n} \frac{1}{Γ(1 + z)} |_{z=α}

= e^{xt} exp[t y c_{z}^{β}] \frac{1}{Γ(1 + z)} |_{z=α}

= e^{xt}W^{(β,α+1)}(t y),  (2.1)

where the Bessel–Wright function W^{(β,α)}(x) [2, 6, 23] reads

W^{(β,α)}(x)= exp[x c_{z}^{β}] \frac{1}{Γ(1 + z)} |_{z=α−1}

= ∑_{r=0}^{∞} \frac{x^{r}}{r!} c_{z}^{β r} \frac{1}{Γ(1 + z)} |_{z=α−1}

= ∑_{r=0}^{∞} \frac{x^{r}}{r! Γ(α + β r)} .

2We did not attempt to demonstrate rigorously the equivalence of these two definitions.

Observe that W^{(β,α)}(x) for β = 1 is equal to [Γ(α)]^{−1}_{0}F_{1} ({− \atop α}; x). Hence Eq. (2.1) can be rewritten as

∑_{n=0}^{∞} \frac{t^{n}}{(1 + α)_{n}} L_{n}^{(α)}(1, −y) = e^{t}_{0}F_{1} ({− \atop 1 + α} ; −ty)
= Γ(1 + α) (y t)^{−α/2} e^{t} J_{α}(2 \sqrt{y t}),

where J_{α}(z) is the Bessel function, see Eq. (9.12.11) on p. 242 of [19] or Eq. (5.11.2.5) for n = 0 on p. 704 of [27].

In the case of ordinary generating function, Eq. (1.20) gives

∑_{n=0}^{∞} t^{n}Λ_{n}^{(α,β)}(x, y) = ∑_{n=0}^{∞}[t (x + y c_{z}^{β})]^{n} \frac{1}{Γ(1 + z)} |_{z=α}

= [\frac{1}{1 − t (x + y c_{z}^{β})}] \frac{1}{Γ(1 + z)} |_{z=α}

= \frac{1}{1 − t x} · [\frac{1}{1 − \frac{t y}{1−t x}c_{z}^{β}} ] \frac{1}{Γ(1 + z)} |_{z=α}

= \frac{1}{1 − t x}E_{β,α+1} (\frac{t y}{1 − t x}),  (2.2)

where E_{β,α}(x) is the two-parameter Mittag-Leffler function [2, 6, 23] defined as

E_{β,α}(x) = [\frac{1}{1 − x c_{z}^{β}} ] \frac{1}{Γ(1 + z)} |_{z=α−1}

= ∑_{r=0}^{∞} x^{r} c_{z}^{β r} \frac{1}{Γ(1 + z)} |_{z=α−1}

= ∑_{r=0}^{∞} \frac{x^{r}}{Γ(α + β r)} ,  (2.3)

The function introduced in Eq. (2.3) for β = 1 is equal to [Γ(α)]^{−1} _{1}F_{1} ({1 \atop α}; x). Thus Eq. (2.2) for β = 1 goes over to Eq. (9.12.12) for γ = 1 on p. 242 of [19], or Eq. (5.11.2.6) for b = 1 on p. 704 of [27], that is,

∑_{n=0}^{∞} \frac{n!}{(1 + α)_{n}} t^{n}L_{n}^{(α)}(1, y)= \frac{1}{1 − t} _{1}F_{1} ({1 \atop 1 + α} ; \frac{t y}{1 − t}).

The ordinary and exponential generating functions of the generalized Laguerre polynomials are accordingly derived as the special case of previous demonstrations; defining^{3} for β = 1:

OG_{1}^{(α,1)}(x, y | t) = ∑_{n=0}^{∞} t^{n}L_{n}^{(α)}(x, y) and _{E}G_{1}^{(α,1)}(x, y | t) = ∑_{n=0}^{∞} \frac{t^{n}}{n!} L_{n}^{(α)}(x, y),

3The subscripts in G’s indicate the degree of lacunarity: 1-no lacunarity, 2-double-lacunary, etc; see also the next paragraph.

we obtain from (1.15), (1.17) and the binomial series expansion:

OG_{1}^{(α,1)}(x, y | t)= Γ(1 + α) [∑_{n=0}^{∞} \frac{(1 + α)_{n}}{n!} (x + y c_{z})^{n}] \frac{1}{Γ(1 + z)} |_{z=α}

= \frac{Γ(1 + α)}{(1 − t x)^{1+α}} [1 − \frac{t y c_{z}}{1 − t x}]^{−1−α} \frac{1}{Γ(1 + z)} |_{z=α}

= \frac{1}{(1 − t x)^{1+α}} [∑_{n=0}^{∞} (\frac{t y}{1 − t x})^{n} \frac{Γ(1 + α + n)}{n!} c_{z}^{n}] \frac{1}{Γ(1 + z)} |_{z=α}

= \frac{1}{(1 − t x)^{1+α}} ∑_{n=0}^{∞} \frac{1}{n!} (\frac{t y}{1 − t x})^{n}

= \frac{1}{(1 − t x)^{1+α}} exp(\frac{t y}{1 − t x}),  (2.4)

which for L_{n}^{(α)}(y) = L_{n}^{(α)}(1, −y) gives the well-known formula (8.975.1) on p. 1002 of [18]. Furthermore, repeating a similar calculation for exponential generating function for integer α = m and β = 1, we have

EG_{1}^{(m,1)}(x, y | t)= [∑_{n=0}^{∞} \frac{(n + m)!}{(n!)^{2}}[t (x + y c_{z})]^{n}]\frac{1}{Γ(1 + z)} |_{z=m}

= [e^{t (x+y c_{z})} ∑_{r=0}^{m} \binom{m}{r} \frac{m!}{r!} [t (x + y c_{z})]^{r}]\frac{1}{Γ(1 + z)} |_{z=m}

= e^{tx} ∑_{r=0}^{m} \binom{m}{r} \frac{m!}{r!} [∑_{s=0}^{r} \binom{r}{s}(t x)^{r−s} (t y)^{s} c_{z}^{s} e^{t y c_{z}}]\frac{1}{Γ(1 + z)} |_{z=m}

= e^{tx} ∑_{r=0}^{m} \binom{m}{r} \frac{m!}{r!} [∑_{s=0}^{r} \binom{r}{s}(t x)^{r−s} (t y)^{s} e^{t y c_{z}}]\frac{1}{Γ(1 + z)} |_{z=m+s}

= e^{tx} ∑_{r=0}^{m} \binom{m}{r} \frac{m!}{r!} ∑_{s=0}^{r} \binom{r}{s}(t x)^{r−s} (t y)^{s} W^{(1,m+s+1)}(t y),  (2.5)

where we used the first Kummer relation for confluent hypergeometric series, see Eq. (9.212) on p. 1023 of [18], in passing from the first to the second line.

3. Lacunary generating functions of Laguerre polynomials

In this paragraph we apply our method to treat various lacunary generating functions for the Laguerre polynomials. The formulas thus obtained appear, for the most part, to be new.

The double-lacunary exponential generating function of ordinary Laguerre polynomials (α = 0)

EG_{2}^{(1,1)}(x, y | t) = ∑_{n=0}^{∞} \frac{t^{n}}{n!} L_{2n}(x, y)  (3.1)

is apparently not known. In the following we obtain the explicit form of the series in Eq. (3.1) in terms of known functions. According to our procedure, we rewrite Eq. (3.1) using Eq. (1.19) as

EG_{2}^{(1,1)}(x, y | t)= [∑_{n=0}^{∞} \frac{t^{n}}{n!} (x + y c_{z})^{2n}] \frac{1}{Γ(1 + z)} |_{z=0}

= e^{t x^{2}}exp[t y^{2} c_{z}^{2} + 2 t x y c_{z}]\frac{1}{Γ(1 + z)} |_{z=0}.  (3.2)

We can provide a definite meaning for the previous expression in terms of known special functions by recalling the expansion

e^{b z^{2}+a z} = ∑_{n=0}^{∞} \frac{z^{n}}{n!} H_{n}^{(2)}(a, b),  (3.3)

with

H_{n}^{(2)}(a, b) = n! ∑_{r=0}^{⌊n/2⌋}\frac{a^{n−2r} b^{r}}{(n − 2r)! r!} ,

where ⌊n⌋ is the floor function. The polynomials H_{n}^{(2)}(a, b) are two-variable Hermite polynomials [1, 7, 13], also defined through the operational rule given in Eq. (1.3). They reduce to the ordinary Hermite polynomials H_{n}(z) through the relation (1.4). The above family of polynomials provides a basis for the definition of the so-called Hermite-based (H-based) functions [8]; for example the Hermite-based (H-based) cylindrical Bessel functions are defined as

HJ_{n}(x, y) = ∑_{r=0}^{∞} (−1)^{r} \frac{H_{n+2r}^{(2)}(x, y)}{2^{n+2r} r! (n + r)!} .

They have been obtained by replacing x^{n+2r} in the relevant series expansion by H_{n+2r}^{(2)}(x, y). According to Eq. (3.3), we obtain

exp[t y^{2} c_{z}^{2} + 2 t x y c_{z}] \frac{1}{Γ(1 + z)} |_{z=0} = [∑_{r=0}^{∞} \frac{c_{z}^{r}}{r!}H_{r}^{(2)}(2 t x y, t y^{2})]\frac{1}{Γ(1 + z)} |_{z=0}

= ∑_{r=0}^{∞} \frac{1}{(r!)^{2}}H_{r}^{(2)}(2 t x y, t y^{2})

= ∑_{r=0}^{∞} \frac{(i \sqrt{t}y)^{r}}{(r!)^{2}}H_{r}(i \sqrt{t}x)  (3.4)

= _{H}C_{0}(−2 t x y, t y^{2}),

where

HC_{α}(x, y) = ∑_{r=0}^{∞} \frac{H_{r}^{(2)}(−x, y)}{r! Γ(1 + α + r)}

is the Hermite-based (H-based) version of the Bessel–Wright function

C_{λ}(x) = ∑_{r=0}^{∞} \frac{(−x)^{r}}{r! Γ(1 + λ + r)} = c_{z}^{λ} exp[−xc_{z}] \frac{1}{Γ(1 + z)} |_{z=0},

which goes over to the Bessel–Wright function W^{β,α}(x) for β = 1 and α = 1 + λ. Thus we get in conclusion

EG_{2}^{(1,1)}(x, y | t) = e^{x^{2} t}_{H}C_{0}(−2 x y t, y^{2} t).

In terms of standard Laguerre polynomials L_{n}(y) = L_{n}(1, −y) Eq. (3.1) can thus be rewritten as

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{2n}(y) = e^{t} ∑_{r=0}^{∞} \frac{(i \sqrt{t}y)^{r}}{(r!)^{2}}H_{r}(i \sqrt{t}),  (3.5)

which constitutes one of key results of the present investigation. The validity of identity (3.5) as well as of further results of this type for other generating functions (see Eqs. (2.5), and (3.7), (3.16), (4.1), (4.3) below) can be independently proven by the substitution t → −t^{2}, followed by coefficient extraction and the use of Eqs. (2.4) and (3.3) [24].

In fact, it turns out that Eq. (3.5)is the special case of a more general relation which extends this result to the generalized Laguerre polynomials L_{2n}^{(α)}(y):

∑_{n=0}^{∞} \frac{(\frac{1}{2})_{n}(−t^{2})^{n}}{(\frac{α}{2} + \frac{1}{2})_{n}(\frac{α}{2} + 1)_{n}} L_{2n}^{(α)}(y)= e^{−t^{2}} ∑_{n=0}^{∞} \frac{(y t)^{n}}{n! (1 + α)_{n}} H_{n}(t).  (3.6)

Using the shift operator c_{z} of Eq. (1.13), the demonstration of Eq. (3.6) works as follows:

∑_{n=0}^{∞} \frac{(−t^{2})^{n}}{n!} \frac{(2n)!}{Γ(1 + α + 2n)} L_{2n}^{(α)}(y)= [∑_{n=0}^{∞} \frac{(−t^{2})^{n}(1 − y c_{z})^{2n}}{n!} ] \frac{1}{Γ(1 + z)} |_{z=α}

= exp[−t^{2} (1 − y c_{z})^{2}] \frac{1}{Γ(1 + z)} |_{z=α}

= e^{−t^{2}} ∑_{n=0}^{∞} \frac{H_{n}(t)}{n!}[t y c_{z}]^{n} \frac{1}{Γ(1 + z)} |_{z=α}

= e^{−t^{2}} ∑_{n=0}^{∞} \frac{(t y)^{n}}{n! Γ(1 + α + n)} H_{n}(t).

Note that Eq. (3.6) for α = 0 and after substitution −t^{2} → t, reproduces Eq. (3.5). We note for completeness another version of the identity Eq. (3.6) rewritten in terms of hypergeometric representations of Laguerre and Hermite polynomials [27]:

∑_{n=0}^{∞} \frac{(1/2)_{n}(1 + α)_{2n}(−t^{2})^{n}}{(2n)! (α/2 + 1/2)_{n}(α/2 + 1)_{n}}_{1}F_{1} ({−2 n \atop 1 + α} ; x)

= e^{−t^{2}} ∑_{n=0}^{∞} \frac{(2 t^{2} x)^{n}}{n! (1 + α)_{n}} _{2}F_{0} ({−n/2, (1 − n)/2 \atop −} ; −\frac{1}{t^{2}}) .

Using the above technique, see Eqs. (3.1)-(3.4), we derive the exponential double-lacunary generating function for the generalized Laguerre polynomials L_{n}^{(α)}(x), for α = m = 1, 2, … and establish that

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{2n}^{(m)}(x, y) = e^{t x^{2}} ∑_{r=0}^{∞} \frac{p_{2m}(r; x, y, t)}{r! (r + 3m)!}H_{r}^{(2)}(2 t x y, t y^{2}),

which for standard generalized Laguerre polynomials L_{n}^{(m)}(y) = L_{n}^{(m)}(1, −y), using Eq. (1.4), has the form

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{2n}^{(m)}(y) = e^{t} ∑_{r=0}^{∞} \frac{p_{2m}(r; 1, −y, t)}{r! (r + 3m)!}(−i \sqrt{t}y)^{r}H_{r}(i \sqrt{t}).  (3.7)

The polynomials p_{2m}(r; x, y, t) are of degree 2m in the variable r, with the coefficients depending on x, y and t. For m = 1, 2, p_{2m}(r; x, y, t) have the explicit form

p_{2}(r; x, y, t)= [1 + 2 x^{2}t] r^{2} + [5 + 4 x y t + 10 x^{2}t] r
+ [6 + 12x^{2}t + 12x y t + 2y^{2}t],  (3.8)
p_{4}(r; x, y, t)= [2 + 10 x^{2}t + 4 x^{4}t^{2}] r^{4}
+ [36 + (180 x^{2} + 20 x y) t + (72 x^{4} + 16 x^{3}y) t^{2}] r^{3}
+ [238 + (10 y^{2} + 1190 x^{2} + 300 x y) t
+ (240 x^{3}y + 24 x^{2}y^{2} + 476 x^{4}) t^{2}] r^{2}
+ [684 + (110 y^{2} + 1480 x y + 3420 x^{2}) t
+ (1184 x^{3}y + 264 x^{2}y^{2} + 16 x y^{3} + 1368 x^{4}) t^{2}] r
+ [720 + (2400 x y + 3600 x^{2} + 300 y^{2}) t
+ (1920 x^{3}y + 96 x y^{3} + 720 x^{2}y^{2} + 1440 x^{4} + 4 y^{4}) t^{2}].  (3.9)

With a moderate effort the polynomials p_{2m}(r; x, y, t) for m > 2 can be also obtained. Explicit forms, various generating functions and combinatorial interpretations of the polynomials (3.8) and (3.9) were recently obtained by Strehl [25, 30].

The current method can be extended to the more general cases like

EG_{2}^{(α,β)}(x, y | t) = ∑_{n=0}^{∞} \frac{t^{n}}{n!}Λ_{2n}^{(α,β)}(x, y),

which, according to our procedure, can be written as

EG_{2}^{(α,β)}(x, y | t)= ∑_{r=0}^{∞} \frac{t^{n}}{n!}(x + y c_{z}^{β})^{2n} \frac{1}{Γ(1 + z)} |_{z=α}

= e^{x^{2} t}exp[2 t x y c_{z}^{β} + t y^{2} c_{z}^{2β}]\frac{1}{Γ(1 + z)} |_{z=α}

= e^{x^{2} t} [∑_{r=0}^{∞} \frac{c_{z}^{βr}}{r!}H_{r}^{(2)}(2 t x y, t y^{2})]\frac{1}{Γ(1 + z)} |_{z=α}

= e^{x^{2}t}_{H}W^{(β,α+1)}(2 t x y, t y^{2})  (3.10)

with

HW^{(α,β)}(x, y) = ∑_{r=0}^{∞} \frac{H_{r}^{(2)}(x, y)}{r! Γ(α r + β)}

being the Hermite-based (H-based) version of the Bessel–Wright function [8].

We give here without demonstration the following two formulas of similar type

∑_{n=0}^{∞} t^{n}L_{2n}(x) = \frac{1}{1 − t} ∑_{r=0}^{∞} \frac{L_{r}^{(r)}(x/2)}{(1/2)_{r}}[−\frac{t x}{2 (1 − t)}]^{r}  (3.11)

and

∑_{n=0}^{∞} t^{n}L_{3n}(x) = \frac{1}{1 − t} ∑_{r=0}^{∞} (−\frac{3 t x}{1 − t})^{r} [∑_{s=0}^{r} \frac{r! (−x)^{s}}{(r − s)! (r + 2s)!} L_{s}^{(s+r)} (x/3)].  (3.12)

In this section we have shown that the tools of employing the symbolic method to derive lacunary generating functions for Laguerre polynomials can easily be applied with a minimum of computation effort.

We now employ the symbolic method to derive old and further new ordinary generating functions for the generalized Laguerre polynomials. We start with a simple observation, namely

c_{z}^{α} exp[b c_{z}^{−1}] \frac{1}{Γ(1 + z)} |_{z=0} = [∑_{n=0}^{∞} \frac{b^{n}}{n!}c_{z}^{α−n} ] \frac{1}{Γ(1 + z)} |_{z=0} = \frac{(1 + b)^{α}}{Γ(1 + α)} .  (3.13)

See also Ref. [4] for a demonstration. We calculate the generating function G_{1}^{(α)}(x, y | t) = ∑ _{n=0}^{∞} t^{n}L_{n}^{(α−n)}(x, y), which after applying Eqs. (1.15), (1.18) and (3.13) can be written as

G_{1}^{(α)}(x, y | t)= Γ(1 + α) [c_{z}^{α} ∑_{n=0}^{∞} \frac{t^{n}}{n!} (\frac{x + y c_{z}}{c_{z}})^{n}] \frac{1}{Γ(1 + z)} |_{z=0}

= Γ(1 + α) e^{t y} c_{z}^{α} exp[t x c_{z}^{−1}] \frac{1}{Γ(1 + z)} |_{z=0} = (1 + t x)^{α} e^{t y},  (3.14)

Formula (3.14) for the ordinary Laguerre polynomials, i. e. for y = 1, is equal to (5.11.4.8) on p. 706 of [27]. In the case of generating function

G_{2}^{(α)}(x, y | t) = ∑_{n=0}^{∞} t^{n}L_{2n}^{(α−2n)}(x, y)  (3.15)

Eqs. (1.15) and (1.17) yield

G_{2}^{(α)}(x, y | t)= Γ(1 + α) [c_{z}^{α} ∑_{n=0}^{∞} \frac{t^{n}}{(2n)!} (\frac{x + y c_{z}}{c_{z}})^{2n}] \frac{1}{Γ(1 + z)} |_{z=0}

= Γ(1 + α) [c_{z}^{α} cosh( \sqrt{t} \frac{x}{c_{z}} + \sqrt{t}y)] \frac{1}{Γ(1 + z)} |_{z=0}

= [cosh( \sqrt{t}y) cosh( \sqrt{t} \frac{x}{c_{z}})+ sinh( \sqrt{t}y) sinh( \sqrt{t} \frac{x}{c_{z}})]\frac{Γ(1 + α)}{Γ(1 + z)} |_{z=α}

= (1 − t x^{2})^{α/2} \{cosh( \sqrt{t}y) cosh[i T_{α}(t, x)] + sinh( \sqrt{t}y) sinh[i T_{α}(t, x)]\}

= (1 − t x^{2})^{α/2} cosh[\sqrt{t}y + i T_{α}(t, x)],  (3.16)

where

T_{α}(t, x) = α arcsin( \frac{\sqrt{t}x}{\sqrt{t x^{2} − 1}}).

Remark: The use of the initial definition of Λ_{n}^{(α)}(x, y), see Eqs. (1.17) and (1.18), in the treatment of G_{1}^{(α)}(x, y | t) and G_{2}^{(α)}(x, y | t) leads to considerable difficulties in obtaining the final results of Eqs. (3.14) and (3.16). One way to avoid these difficulties is to adopt a modified (albeit strictly equivalent) definition of Λ_{n}^{(α−n)}(x, y) in the form

Λ_{n}^{(α−n)}(x, y) = c_{z}^{α−n}(x + y c_{z})^{n} \frac{1}{Γ(1 + z)} |_{z=0}.

Actually this last definition was used in deriving Eqs. (3.14) and (3.16), whereas the previous one was used everywhere else.

4. Further developments

The method we have developed so far can be extended to obtain even slightly more complicated expressions like

EG_{2,ℓ}^{(1,1)}(x, y | t) = ∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{2n+ℓ}(x, y) and _{E}G_{3,ℓ}^{(1,1)}(x, y | t) = ∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{3n+ℓ}(x, y),

for l = 1, 2, . . .. Note that corresponding formulas for the Hermite polynomials were obtained in [15]. Applying Eqs. (1.14), (3.2), and (3.3) for the double lacunary generating function of Laguerre polynomials, we have

EG_{2,ℓ}^{(1,1)}(x, y | t)= e^{t x^{2}} [(x + c_{z}y)^{ℓ} ∑_{r=0}^{∞} \frac{c_{z}^{r}}{r!} H_{r}^{(2)}(2 t x y, t y^{2})]\frac{1}{Γ(1 + z)} |_{z=0}

= e^{t x^{2}} ∑_{s=0}^{ℓ} \binom{ℓ}{s}x^{ℓ−s}y^{s} _{H}C_{s}(−2 t x y, t y^{2}),

which, for L_{n}(y) = L_{n}(1, −y),can be written in terms of standard generalized Laguerre and Hermite polynomials as follows:

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{2n+ℓ}(x)= e^{t} ∑_{r=0}^{∞} \frac{(−i \sqrt{t})^{r}}{r!}H_{r}(i \sqrt{t}x) ∑_{s=0}^{ℓ} \binom{ℓ}{s}\frac{x^{l−s}}{(r + s)!}

= e^{t}ℓ! ∑_{r=0}^{∞} \frac{(i \sqrt{t}x)^{r}}{r! (ℓ + r)!} L_{ℓ}^{(r)}(x) H_{r}(i \sqrt{t}).  (4.1)

Using the analogous procedure, we obtain also_{E}G_{3,ℓ}^{(1,1)}(x, y | t) in the form

EG_{3,ℓ}^{(1,1)}(x, y | t)= e^{t x^{3}} [(x + y c_{z})^{ℓ} ∑_{n=0}^{∞} \frac{c_{z}^{n}}{n!} H_{n}^{(3)}(3 t x^{2} y, 3 t x y^{2}, t y^{3})] \frac{1}{Γ(1 + z)} |_{z=0}

= e^{t x^{3}} ∑_{s=0}^{ℓ} \binom{ℓ}{s}y^{s}x^{ℓ−s}_{H}C_{s}^{(3)}(3 t x^{2} y, 3 t x y^{2}, t y^{3}),  (4.2)

where

HC_{s}^{(3)}(x, y, z) = ∑_{r=0}^{∞} \frac{H_{r}^{(3)}(x, y, z)}{r! (r + s)!}

is a third order Hermite-based (H-based) Tricomi function [8], with

H_{n}^{(3)}(x, y, z) = n! ∑_{r=0}^{⌊n/3⌋}\frac{z^{r} H_{n−3r}^{(2)}(x, y)}{r! (n − 3r)!}

being a third order three-variable Hermite polynomial, with the generating function [10]

∑_{n=0}^{∞} \frac{t^{n}}{n!}H_{n}^{(3)}(x, y, z) = e^{t x+t^{2} y+t^{3} z}.

Eq. (4.2) expressed via standard Laguerre and Hermite polynomials is given as

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{3n+ℓ}(x) = e^{t}ℓ! ∑_{n=0}^{∞} \frac{L_{ℓ}^{(n)}(x)}{(n + ℓ)!} ∑_{r=0}^{⌊n/3⌋}\frac{(−t x^{3})^{r}(i \sqrt{3 t}x)^{n−3r}}{r! (n − 3 r)!}H_{n−3r} (i \frac{\sqrt{3 t}}{2}). (4.3)

In the case of the generalized Laguerre polynomials for α = 1 and ℓ = 0, L_{3n}^{(1)}(1, −y) = L_{3n}^{(1)}(y), the generating function is given as

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{3n}^{(1)}(y) = e^{t} ∑_{r=0}^{∞} \frac{q_{3}(r; y, t)}{r! (r + 4)!}H_{r}^{(3)}(−3 t y, 3 t y^{2}, −t y^{3}),

= e^{t} ∑_{r=0}^{∞} \frac{y^{r} q_{3}(r; y, t)}{(r + 4)!}∑_{s=0}^{⌊r/3⌋}\frac{(−t)^{s}(i \sqrt{3 t})^{r−3 s}}{s! (r − 3s)!} H_{r−3s} (i \frac{\sqrt{3 t}}{2})  (4.4)

with

q_{3}(r; y, t)= [1 + 3 t] r^{3} + [9 + 27 t − 9 y t] r^{2} + [26 + 78 t − 63 y t + 9 y^{2} t] r
+ [24 + 72 t − 108 y t + 36y^{2} t − 3 y^{3} t],

which is a third order polynomial in r, with the coefficients depending on y and t. Its closed form and the analytical and combinatorial properties were worked out by Strehl [25, 30].

The extension to the case

EG_{m,ℓ}^{(1,1)}(x, y | t) = ∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{mn+ℓ}(x, y)

can be straightforwardly accomplished within the present framework and reads

EG_{m,ℓ}^{(1,1)}(x, y | t)= e^{t y^{m}}[(x + y c_{z})^{ℓ} ∑_{r=0}^{∞} \frac{c_{z}^{r}}{r!} H_{r}^{(m)}(−α_{1}, α_{2}, …, (−1)^{m}α_{m})] \frac{1}{Γ(1 + z)} |_{z=0}

= e^{t x^{m}} ∑_{s=0}^{ℓ} \binom{ℓ}{s}x^{ℓ−s} y^{s} _{H}C_{s}^{(m)}(α_{1}, α_{2}, …, α_{m}),

α_{p} = \binom{m}{p}x^{m−p} (−y)^{p} t, p = 1, 2, …m,  (4.5)

where _{H}C_{n}^{(m)}(α_{1}, α_{2}, …, α_{m}) is an m-th order Hermite-based Tricomi function, defining as basis Hermite polynomials H_{n}^{(m)}(−α_{1}, α_{2}, …, (−1)^{m}α_{m}) in m variables, as specified by the generating function [10]

∑_{n=0}^{∞} \frac{t^{n}}{n!}H_{n}^{(m)}(x_{1}, x_{2}, …, x_{m})= exp (∑_{s=1}^{m} x_{s} t^{s}).

Let us finally consider the bilateral generating function

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{n}(x, y) L_{n}(v, u) =

∑_{n=0}^{∞} \frac{t^{n}}{n!} [(x + y c_{z})^{n} \frac{1}{Γ(1 + z)} |_{z=0}] [(v + u cz̃)^{n} \frac{1}{Γ(1 + z̃)} |_{z̃=0}].

We obtain

∑_{n=0}^{∞} \frac{t^{n}}{n!}L_{n}(x, y) L_{n}(v, u)= e^{t x v}[e^{−t y v c_{z}} \frac{1}{Γ(1 + z)} |_{z=0}] [e^{−t x u cz̃} \frac{1}{Γ(1 + z̃)} |_{z̃=0}]

× (e^{t y u c_{z}cz̃} \frac{1}{Γ(1 + z)} |_{z=0} \frac{1}{Γ(1 + z̃)} |_{z̃=0})

= e^{t x v t}_{H}C_{0,0}(t y v, t x u | t y u),

where

HC_{0,0}(x, y | τ) = ∑_{r,s,k=0}^{∞} \frac{x^{r} y^{s} τ^{k}}{r! s! k! (r + k)! (s + k)!} .

Closely related considerations devoted to combinatorics of Laguerre polynomials are developed in [14].

Using the operational method, we can also obtain other interesting identities for the lacunary generating functions of the generalized Laguerre polynomials. These formulas are listed below:

∑_{n=0}^{∞} \frac{(\frac{1}{2})_{n}t^{n}}{(1 + \frac{α}{2})_{n}}L_{2n}^{(α)}(x)= (1 − t)^{−(1+α/2)} ∑_{r=0}^{∞} \frac{L_{r}^{(r+α)} (x/2)}{(1 + α/2)_{r}} (−\frac{t x}{2 (1 − t)})^{r}

= \frac{1}{\sqrt{1 − t}} ( \sqrt{t} \frac{x}{2})^{−α/2} exp(−\frac{t x}{1 − t}) I_{α/2} ( \sqrt{t} \frac{x}{1 − t}),  (4.6)

obtained with formula (5.11.4.12) of [27], which for α = 2m gives

∑_{n=0}^{∞} \frac{(\frac{1}{2})_{n} t^{n}}{(1 + m)_{n}}L_{2n}^{(2m)}(x)= \frac{1}{\sqrt{1 − t}} ( \sqrt{t} \frac{x}{2})^{−m} exp(−\frac{t x}{1 − t}) I_{m}( \sqrt{t} \frac{x}{1 − t}),  (4.7)

where m = 0, 1, 2, …and I_{m}(z) is the modified Bessel function;

∑_{n=0}^{∞} \frac{(1/3)_{n} (2/3)_{n}t^{n}}{(1 + α/3)_{n} (2/3 + α/3)_{n}} L_{3n}^{(α)}(x)

= (1 − t)^{−(1+α)/3} ∑_{r=0}^{∞} \frac{Γ(1 + α + 3r)}{(1 + α/3)_{r} (2/3+ α/3)_{r}} (−\frac{t x}{9 (1 − t)})^{r}

× [∑_{s=0}^{r} \frac{(−x)^{s} L_{s}^{(α+r+s)} (x/3)}{(r − s)! Γ(1 + α + r + 2s)} ]. (4.8)

Eq. (4.7) corrects the formula (5.11.2.10), p. 704 of [27].

Let us now derive Eq. (4.6). Following Eqs. (1.15), (1.17)and the Gauss–Legendre formula for Γ(2 z), we get

∑_{n=0}^{∞} \frac{(1/2)_{n}t^{n}}{(1 + α/2)_{n}}L_{2n}^{(α)}(x) = ∑_{n=0}^{∞} \frac{(1/2)_{n} t^{n}}{(2n)!} \frac{Γ(1 + α + 2n)}{(1 + α/2)_{n}} [c_{z}^{α} (1 − x c_{z})^{2n}] \frac{1}{Γ(1 + z)} |_{z=0}

= Γ(1 + α) [c_{z}^{α} ∑_{n=0}^{∞} \frac{(1/2 + α/2)_{n}}{n!}[t (1 − x c_{z})^{2}]^{n}] \frac{1}{Γ(1 + z)} |_{z=0}

= \frac{Γ(1 + α)}{(1 − t)^{(1+α)/2}} [1 + c_{z}(\frac{2 t x − t x^{2} c_{z}}{1 − t})]^{−(1+α)/2} \frac{1}{Γ(1 + z)} |_{z=α}

= \frac{Γ(1 + α)}{(1 − t)^{(1+α)/2}} ∑_{r=0}^{∞} \frac{(−1)^{r}}{r!} (\frac{1 + α}{2} )_{r} c_{z}^{r}(\frac{2 t x − t x^{2} c_{z}}{1 − t})^{r} \frac{1}{Γ(1 + z)} |_{z=α}

= \frac{Γ(1 + α)}{(1 − t)^{(1+α)/2}} ∑_{r=0}^{∞} \frac{(−1)^{r}}{r!} (\frac{1 + α}{2} )_{r} Λ_{r}^{(r+α)} (\frac{t x^{2}}{1 − t}, \frac{2 t x}{1 − t})

= (1 − t)^{−(1+α)/2} ∑_{r=0}^{∞} \frac{L_{r}^{(r+α)}(x/2)}{(1 + α/2)_{r}}(−\frac{t x}{2 (1 − t)})^{r} .

Eq. (4.7) comes from Eq. (4.6) for α = 2m and using formula (5.11.4.12) on p. 706 of [27]. An alternative demonstration of Eq. (4.7) can be carried through using mixed bilateral generating function of Laguerre and Gegenbauer polynomials, see Appendix A.

5. Concluding Remarks

Having obtained the summation formulas Eqs. (4.6) and (4.7), we may write down the right hand sides of Eqs. (1.11) and (1.12) in explicit form. For α = 0, the initial condition for Eqs. (1.9) and (1.10), f (x) = I_{0}(βx), β > 0, evolves with time according to

exp[−t D_{x}^{(0)}] I_{0}(βx) = \frac{1}{\sqrt{1 − β^{2} t^{2}}} exp(−\frac{β^{2} t x}{1 − β^{2} t^{2}}) I_{0} (\frac{β x}{1 − β^{2} t^{2}}),  (5.1)

≡ P_{0}(t, x) I_{0} (\frac{β x}{1 − β^{2} t^{2}}), (β t)^{2} < 1.  (5.2)

Furthermore, from Eq. (5.1) one obtains

exp[−t D_{x}^{(α)}] x^{−α/2} I_{α/2}(β x)

= \frac{(1 − β^{2} t^{2})^{−(1+α)/2}}{Γ(1 + α/2)} exp(−\frac{β^{2} t x}{1 − β^{2} t^{2}}) (\frac{x}{1 − β^{2} t^{2}})^{−α/2} I_{α/2} (\frac{β x}{1 − β^{2} t^{2}}),

≡ P_{α}(t, x) (\frac{x}{1 − β^{2} t^{2}})^{−α/2} I_{α/2} (\frac{β x}{1 − β^{2} t^{2}}), (β t)^{2} < 1. (5.3)

In Eq. (5.3) we defined the “prefunction” P_{α}(t, x) > 0 satisfying P_{α}(0, x) = 1. Eqs. (5.1) and (5.2) illustrate the scaling property of the time evolution: we show that special solutions exists which, up to the “prefunction” P_{α}(t, x), consist in rescaling of the argument of the initial condition f (x) according to x → βx/(1 − β^{2} t^{2}). This feature is characteristic for a number of examples of the so-called generalized Glaisher relations, see [16] and references therein. An elementary example of a relation of this type is given in Eq. (1.7). Many similar examples can be worked out by choosing different forms of f (x) and different values of m in Eq. (4.7).

Before concluding this paper it is worth presenting some further comments to reconcile the present results with previous approaches based on the monomiality principle [7, 11, 12, 13, 22]. Within that context the so called Laguerre derivative D_{x}^{(0)} has been introduced in Eq. (1.1), so that

D_{x}^{(0)}L_{n}(x, y) = n L_{n−1}(x, y).

The Laguerre derivative in differential terms is defined as D_{x}^{(0)} = \frac{d}{dx}x\frac{d}{dx}. As a further example of application we consider the action of the exponential containing the Laguerre derivative acting on the Tricomi function C_{0}(x) = I_{0}(2 \sqrt{x}), namely

exp[y D_{x}^{(0)}] C_{0}(x) = …= e^{y}C_{0}(x),

in agreement with the fact that C_{0}(x) is an eigenfunction of the Laguerre derivative [7, 13] with the eigenvalue equals to one. An alternative operator definition of the Laguerre derivative is given in Appendix B, where also some of its consequences are explored. The formal procedure we have developed can be pushed even further.

The use of the following identity

K^{x \frac{d}{dx}}f (x) = f (Kx).

and the definition of the 0-th order cylindrical Bessel function as a pseudo-Gaussian [8], namely

J_{0}(x) = e^{−c_{z}(x/2)^{2}} \frac{1}{Γ(1 + z)} |_{z=0},

allow for the derivation of the following identity

c_{z}^{− \frac{1}{2}x \frac{d}{dx}}J_{0}(x) = e^{−(x/2)^{2}},

which, in terms of integral transforms, can be interpreted as a kind of Borel transform [12]

c_{z}^{− \frac{1}{2}x \frac{d}{dx}}J_{0}(x) = ∫ _{0}^{∞} e^{−s} J_{0}( \sqrt{s}x)ds.

Finally let us note that, having expressed the cylindrical Bessel functions in terms of Gaussian, it is also possible to “reduce” a Gaussian to a Lorentzian, according to the identity

e^{−x^{2}} = \frac{1}{1 + x^{2} c_{z}} \frac{1}{Γ(1 + z)} |_{z=0}.

According to this last identity, we can write the relevant integral as

∫ _{0}^{x}e^{−ξ^{2}} dξ = ∫ _{0}^{x} \frac{dξ}{1 + c_{z}ξ^{2}} \frac{1}{Γ(1 + z)} |_{z=0} = [c_{z}^{−1/2} arctan( \sqrt{c_{z}}x)] \frac{1}{Γ(1 + z)} |_{z=0}.

Here we give for completeness the list of new closed form expressions obtained in the present investigation. These are equations: (2.1), (2.2), (2.5), (3.5), (3.6), (3.7), (3.10), (3.11), (3.12), (3.16), (4.1), (4.3), (4.4), (4.5), (4.6), (4.7), and (4.8). The methods we have illustrated in this paper appear fairly flexible and amenable for further implementations, as will be shown in a future investigation.

6. Acknowledgments

We thank Volker Strehl, Dominique Foata and Alin Bostan for important discussions.

A short time after submission of the original version of the present article, Volker Strehl informed the authors that he has proven by purely combinatorial means virtually all the new summation formulas reported here. He has also provided explicit forms of polynomials p_{2}(r; x, y, t), see Eq. (3.8), p_{4}(r; x, y, t), see Eq. (3.9), and q_{3}(r; x, t) on page 12, thereby closing the remaining gaps. In addition he has obtained a number of new formulas.

We believe that Strehl’s significant results constitute a very substantial step towards establishing new links between the umbral and combinatorial methods. We thank Dr. Strehl for encouragement and for ample correspondence on these subjects. We thank both referees for constructive remarks.

This work has been supported by Agence Nationale de la Recherche (Paris, France) under Program PHYSCOMB No. ANR-08-BLAN-0243-2 and PHC Polonium, Campus France, project no. 28837QA. KG thanks for the support from MNiSW, Warsaw, Poland, “Iuventus Plus 2015-2016”, program no. IP2014 013073.

Appendix A.

This appendix is entirely based on the material kindly communicated to the authors by Strehl [26].

The identity Eq. (4.7),

∑_{n=0}^{∞} \frac{(1/2)_{n} t^{n}}{(1 + m)_{n}}L_{2n}^{(2m)}(x) = \frac{1}{\sqrt{1 − t}} ( \frac{\sqrt{t}x}{2})^{−m} exp(\frac{−t x}{1 − t}) I_{m}(\frac{\sqrt{t}x}{1 − t}),  (A.1)

for 0 < t < 1 and m = 0, 1, …, will be shown to follow from Eq. (1.20), § 144, p. 281 of [28], namely from the following mixed bilateral generating function of products of Laguerre and Gegenbauer polynomials:

∑_{n=0}^{∞} \frac{n! t^{n}}{(γ)_{n}}L_{n}^{(γ−1)}(y) C_{n}^{γ/2}(x) = ρ^{−γ} exp(\frac{−t (x − t) y}{ρ^{2}}) _{0}F_{1} ({− \atop (1 + γ)/2}; \frac{t^{2} y^{2}}{4 ρ^{4}}(x^{2} − 1)) , (A.2)

where C_{n}^{γ/2}(x) is a Gegenbauer polynomials and ρ = \sqrt{1 − 2xt + t^{2}}. (As apparently first shown by Weisner [31], Eq. (A.2) can be derived using group-theoretic methods.)

On the left hand side of Eq. (A.2) we set x = 0 and observe that C_{2n}^{ν}(0) = (−1)^{n}(ν)_{n}/n! and C_{2n+1}^{ν}(0) = 0, see p. 732 of [27]; then in the sum, only even terms in n survive. Furthermore, upon replacing t → i \sqrt{t}in this sum we end up with

∑_{n=0}^{∞} \frac{(2 n)!}{(γ)_{2n}} \frac{(−1)^{n}}{n!} L_{2n}^{(γ−1)}(y) (i \sqrt{t})^{2n}(γ/2)_{n} = ∑_{n=0}^{∞} \frac{(1/2)_{n} t^{n}}{(1/2 + γ/2)_{n}}L_{2n}^{(γ−1)}(y),  (A.3)

where we have used the relation

\frac{(2 n)!}{(γ)_{2n}} \frac{(γ/2)_{n}}{n!} = \begin{matrix}(1/2) & n \\ ((1 + γ)/2) & n \end{matrix}.

Observe that Eq. (A.3) can be identified with the left hand side of (A.1) by setting m = (γ − 1)/2. We apply now the above substitutions (x = 0 and t → i \sqrt{t}) to the right hand side of Eq. (A.2) which then becomes

(1 − t)^{−γ/2} exp(−\frac{t y}{1 − t}) _{0}F_{1} ({− \atop (1 + γ)/2}; (\frac{\sqrt{t}y}{2 (1 − t)})^{2}).  (A.4)

In the final step, the use in Eq. (A.4) of the hypergeometric representation of Bessel functions I_{m}(z), see p. 729 of [27],

I_{m}(z) = \frac{1}{Γ(1 + m)} (\frac{z}{2})^{m}_{0}F_{1} ({− \atop 1 + m}; \frac{z^{2}}{4}),

accomplishes the proof of Eqs. (4.7) and (A.1).

Appendix B.

In the main body of the paper we have introduced the concept of Laguerre derivative, which is a useful tool to deal with special polynomials of the Laguerre type. More generally it provides a key operator to further simplify many of the computational tasks we have dealt with, provided that we embed its definition within the envisaged umbral restyling.

We note indeed that the use of this operator allows the following alternative definition:

LD_{x} = \frac{d}{dX}, where X = x c_{z},

as a consequence of the fact that

\frac{dX^{n}}{dX} = n X^{n−1}.

If we now use the following special notation

exp[λ_{L}D_{x}] = exp[\frac{λ}{c_{z}} \frac{d}{dx}],

by keeping separate the actions of the derivative and umbral operators, we find, e. g.,

exp[λ_{L}D_{x}] X^{n}= exp[\frac{λ}{c_{z}} \frac{d}{dx}]c_{z}^{n}x^{n} \frac{1}{Γ(1 + z)} |_{z=0}

= c_{z}^{n}(x + λ c_{z}^{−1})^{n} \frac{1}{Γ(1 + z)} |_{z=0}

= (λ + x c_{z})^{n} \frac{1}{Γ(1 + z)} |_{z=0}

= L_{n}(λ, x) = λ^{n}L_{n}(−x/λ) = exp[λ D_{x}^{(0)}] \frac{x^{n}}{n!},

or

exp[λ_{L}D_{x}]e^{a X} = exp[\frac{λ}{c_{z}} \frac{d}{dx}] exp[a x c_{z}] \frac{1}{Γ(1 + z)} |_{z=0}

= exp[a c_{z}(x + λ c_{z}^{−1})] \frac{1}{Γ(1 + z)} |_{z=0}

= e^{a λ} exp[a x c_{z}] \frac{1}{Γ(1 + z)} |_{z=0} = e^{a λ} I_{0}(2 \sqrt{a x}).

Observe also that

exp[λ_{L}D_{x}] e^{a X} = exp[λ_{L}D_{x}] ∑_{r=0}^{∞} \frac{(a X)^{r}}{r!}

= ∑_{r=0}^{∞} \frac{a^{r}}{r!} exp[λ_{L}D_{x}] X^{r}

= ∑_{r=0}^{∞} \frac{a^{r}}{r!} exp[λ D_{x}^{(0)}] \frac{x^{r}}{r!}

= exp[λ D_{x}^{(0)}]∑_{r=0}^{∞} \frac{(a x)^{r}}{(r!)^{2}} = exp[λ D_{x}^{(0)}] I_{0}(2 \sqrt{ax}).

The same definition can be used in an even more raffishly way by noting that

exp[λ_{L}D_{x}] e^{−x} = exp[\frac{λ}{c_{z}} \frac{d}{dx}] \frac{1}{1 + xc_{z}} \frac{1}{Γ(1 + z)} |_{z=0}

= \frac{1}{1 + c_{z}(x + λ c_{z}^{−1})} \frac{1}{Γ(1 + z)} |_{z=0}

= \frac{1}{1 + λ} \frac{1}{1 + (x c_{z})/(1 + λ)} \frac{1}{Γ(1 + z)} |_{z=0}

= \frac{1}{1 + λ} exp(−\frac{x}{1 + λ} ) = exp[λ D_{x}^{(0)}]e^{−x} .

References

[1] Appell P. and Kampé de Fériet J., “Fonctions Hypergéométriques et Hypersphériques: Polynômes d’Hermite” (Gauthier–Villars, Paris, 1926).

[2] Babusci D. and Dattoli G., On evaluation of integrals involving Bessel functions, preprint

arχiv:1111.0881v1.

[3] Babusci D., Dattoli G., Duchamp G. H. E., Górska K., Penson K. A., Definite integrals and operational methods, Appl. Math. Comp. 219 (2012) 3017.

[4] Babusci D., Dattoli G., Górska K., and Penson K. A., Symbolic methods for the evaluation of sum rules of Bessel functions, J. Math. Phys. 54 (2013) 073501.

[5] Babusci D., Dattoli G., Górska K., Penson K. A., and Duchamp G. H. E., The Ramanujan master theorem and its implications for special functions, Appl. Math. Comp. 218 (2012) 11466.

[6] Babusci D., Dattoli G., and Górska K., On Mittag-Leffler function and associated polynomials, preprint

arχiv:1206.3495v1.

[7] Cocolicchio D., Dattoli G., and Srivastava H. M., (eds.) “Advanced special functions and applications”, Proceedings of the Workshop, Melfi (PZ), Italy, 9–12 May 1999 (Aracne Editrice, Rome).

[8] Dattoli G., Advanced Special Functions and Integration Methods, Proceedings of the Workshop, Melfi (PZ), Italy, 18–23 June 2000.

[9] Dattoli G., He M. X., and Ricci P. E., Eigenfunctions of Laguerre-operators and generalized evolution problems, Math. Comput. Model. 42 (2005) 1263.

[10] Dattoli G., Lorenzutta S., Cesarano C., and Ricci P. E., Second level exponentials and families of Appell polynomials, Int. Trans. Spec. Fun. 13 (2002) 521.

[11] Dattoli G., Mancho A., and Torre A., The generalized Laguerre polynomials, the associated Bessel functions and applications to propagation problems, Radiat. Phys. Chem. 53 (1998) 391.

[12] Dattoli G., Martinelli M. R., and Ricci P. E., On new families of integral transforms for the solution of partial differential equations, Int. Trans. Spec. Fun. 16 (2005) 661.

[13] Dattoli G., Ricci P. E., and Khomasuridze I., Operational methods, special polynomial and functions and solution of partial differential equations, Int. Trans. Spec. Fun. 15 (2004) 309.

[14] Foata D. and Strehl V., Combinatorics of Laguerre polynomials, in “Enumeration and Design”, 123–140 (Academic Press, Toronto, 1984).

[15] Gessel I. M. and Jayawant P., A triple lacunary generating function for Hermite polynomials, Electronic J. Comb. 12 (2005) R30.

[16] Górska K., Horzela A., Penson K. A., and Dattoli G., The Higher-order heat-type equations via signed Lévy stable and generalized Airy functions, J. Phys. A: Math. Theor. 46 (2013) 425001.

[17] Górska K., Penson K. A., and Dattoli G., Lacunary generating functions for Legendre and Chebyshev polynomials, unpublished.

[18] Gradshteyn I. S. and Ryzhik I. M., “Table of Integrals, Series, and Products”, 7th Edition, (Elsevier, New York, 2007).

[19] Koekoek R., Lesky P. A., and Swarttouw R. F. , “Hypergeometric Orthogonal Polynomials and Their q-Analogues” (Springer, Heidelberg, 2010).

[20] Kung J. P. S., Rota G.-C., and Yan C. H., “Combinatorics, the Rota Way”, (Cambridge University Press, Cambridge, 2009).

[21] Louck J. D., “Unitary Symmetry and Combinatorics”, (World Scientific, Singapore, 2008).

[22] Penson K. A., Blasiak P., Horzela A., Duchamp G. H. E., and Solomon A. I., Laguerre-type derivatives: Dobiński relations and combinatorial identities, J. Math. Phys. 50 (2009) 083512.

[23] Podlubny I., “Fractional Differential Equations” (Academic Press, San Diego, 1999).

[24] Private communication from A. Bostan, April 15, 2011.

[25] Private communications from V. Strehl, August 7–21, 2013.

[26] Private communication from V. Strehl, February 28, 2014.

[27] Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., “Integrals and Series. Special Functions”, vol. 2., (Gordon and Breach, Amsterdam, 1998).

[28] Rainville E. D., “Special Functions” (Chelsea Publ. Comp., 1960).

[29] Roman S., “The Umbral Calculus”, (Academic Press, New York, 1984).

[30] Strehl V., Lacunary Laguerre Series from a Combinatorial Perspective, Séminaire Lotharingien de Combinatoire 76 (2017), Article B76c.

[31] Weisner L., Group theoretic origins of certain generating functions, Pacific J. Math. 5 (1954) 1033.

INFN - Laboratori Nazionali di Frascati, via E. Fermi, 40, IT 00044 Frascati (Roma), Italy

  E-mail address: danilo.babusci@lnf.infn.it

ENEA - Centro Ricerche Frascati, via E. Fermi, 45, IT 00044 Frascati (Roma), Italy

  E-mail address: dattoli@frascati.enea.it

H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL 31-342 Kraków, Poland

  E-mail address: katarzyna.gorska@ifj.edu.pl

Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne Universités, Université Pierre et Marie Curie, CNRS UMR 7600 Tour 13 - 5ième ét., B.C. 121, 4 pl. Jussieu, F 75252 Paris Cedex 05, France

  E-mail address: penson@lptl.jussieu.fr