Identities involving 3-variable Hermite polynomials arising from umbral method

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R E S E A R C H

Open Access

Identities involving 3-variable Hermite

polynomials arising from umbral method

Nusrat Raza

1

_{, Umme Zainab}

2

_{, Serkan Araci}

3*

_{and Ayhan Esi}

4

*_{Correspondence:} serkan.araci@hku.edu.tr

3_{Department of Economics, Faculty}

of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410, Gaziantep, Turkey

Full list of author information is available at the end of the article

Abstract

In this paper, we employ an umbral method to reformulate the 3-variable Hermite polynomials and introduce the 4-parameter 3-variable Hermite polynomials. We also obtain some new properties for these polynomials. Moreover, some special cases are discussed and some concluding remarks are also given.

MSC: 05A40; 33C45

Keywords: Umbral method (umbra); 3-variable generalised Hermite polynomials; 4-parameter 3-variable Hermite polynomials; Generating function

1 Introduction

The multi-variable Hermite polynomials have been used in the analysis of charged-beam transport problems in classical mechanics as well as in the formulation of quantum-phase-space mechanics. Umbral methods have been largely exploited to study the properties of the Hermite polynomials. Recently Dattoli et al. applied the method of umbral to obtain certain results for the Hermite polynomials [8]. The study of umbral formalism provides a fairly helpful tool in many topics of practical nature concerning physics of free electron laser. In this paper, we extend the umbral treatment of the Hermite polynomials from two variables to three variables.

We begin with some umbral results on the 2-variable Hermite polynomials (2VHP)

Hn(x, y). We recall that 2VHP Hn(x, y) are deﬁned by means of the following generating function and series deﬁnition [2]:

∞ n=0 Hn(x, y) tn n!= e xt+yt2 ₍₁₎ and Hn(x, y) = n! [n2] k=0 xn–2k_yk k!(n – 2k)!, (2) respectively.

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The boundary conditions for 2VHP Hn(x, y) are as follows [8,21]: Hn(x, 0) = xn (3) and Hn(0, y) = n! yn2 (n₂+ 1) cosnπ 2 , (4) respectively.

In this paper, we employ the umbral method to the 3-variable Hermite polynomials. Also, we exploit the umbral method to obtain several extensions of the 3-variable Hermite polynomials. Recently, Dattoli et al. gave an umbral method for 2VHP Hn(x, y), which plays an important role in the ﬁeld of special functions and applied mathematics to obtain all the relevant properties of the other special polynomials as well as special functions [8].

In [8], Dattoli considered the idea of umbra, denoted by ˆby, for 2VHP Hn(x, y) as follows: ˆbr yφ0= yr2_r_! (₂r+ 1) cosrπ₂ (φ0= 0), (5)

where φ0 is known as polynomial vacuum and ˆby acting on the state φ0 yields 2VHP Hn(x, y).

The exponential of umbra ˆbyis particulary important to derive the generating functions for 2VHP Hn(x, y). The exponential of umbra ˆbyis as follows [8]:

eˆbyt_φ 0= eyt

2

. (6)

In view of equation (5), 2VHP Hn(x, y) can be reduced binomially as follows:

Hn(x, y) = (x + ˆby)nφ0, (7)

see [8].

Dattoli [8] introduced the 2-parameter 2-variable Hermite polynomials 2P2VHP Hn(x, y|β, α): Hn(x, y|β, α) = ˆbβy x+ ˆbα y n φ0. (8)

The generating function of 2P2VHP Hn(x, y|β, α) is as follows: ∞ n=0 Hn(x, y|β, α) tn n!= e xt_yβ₂_e(α,β)_yα₂_t_, ₍₉₎ where e(α,β)(x) = ∞ r=0 (αr + β + 1)xr (αr₂+β+ 1)r! cosαr₂+ βπ, (10)

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see [8], which is a generalisation of the exponential function. It is worthy to note that

e(1,0)(x) = ex

2

. (11)

Now, we recall that the 3-variable Hermite polynomials (3VHP) Hn(x, y, z) are deﬁned by means of the following generating function and series deﬁnition [6]:

∞ n=0 tn n!Hn(x, y, z) = exp xt+ yt2+ zt3 (12) and Hn(x, y, z) = n! [n₃] k=0 Hn–3k(x, y)zk k!(n – 3k)! , (13) respectively.

The operational deﬁnition of 3VHP Hn(x, y, z) is as follows [6]: Hn(x, y, z) = ezD 3 x+yD2x_xn_, ₍₁₄₎ where Dx:= d dx.

The Gould–Hopper polynomials (GHP) Hn(m)(x, y) are deﬁned by means of the following generating function and series deﬁnition [13]:

∞ n=0 H_n(m)(x, y)t n n! = e xt+ytm ₍₁₅₎ and H_n(m)(x, y) = n! [_mn] r=0 xn–mr_yr r!(n – mr)!, (16) respectively. Since mˆbryφ0= ymr_r_! (_mr + 1)Am,r, (17) Am,r= ⎧ ⎨ ⎩ 1 r= mp, p∈ N, 0, otherwise. (18)

Dattoli [8] deﬁned GHP Hn(m)(x, y) in terms of the nth power of the binomial given by

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The 3-variable generalised Hermite polynomials (3VgHP) Hn(s,m)(x, y, z) are deﬁned by means of the following generating function [11]:

∞ n=0 H_n(s,m)(x, y, z)t n n!= e (xt+ytm_+zts₎ (20) and equivalently by H_n(s,m)(x, y, z) = n! [n_s] r=0 Hn(m)–sr(x, y)zr (n – sr)!r! . (21)

The operational deﬁnition of 3VgHP Hn(s,m)(x, y, z) are as follows [11]:

H_n(s,m)(x, y, z) = expzDs_x+ yDm_xxn (22) and

H_n(s,m)(x, y, z) = expzDs_xH_n(m)(x, y). (23) In this paper, motivated by the work of Dattoli on the umbral behaviour of the Hermite polynomials [8,10], we extend the umbral formalism to the 3-variable Hermite polyno-mials. In Sect.2, we deﬁne an umbra for the 3-variable Hermite polynomials and obtain umbral deﬁnition for 3-variable Hermite polynomials and 3-variable generalised Hermite polynomials. In Sect.3, we introduce an extension of 3-variable Hermite polynomials to 4-parameter 3-variable Hermite polynomials by using the umbral formalism and estab-lish certain results involving these polynomials. In Sect.4, we discuss some special cases of 4-parameter 3-variable Hermite polynomials. Some concluding remarks are given in Sect.5.

2 Umbra and 3-variable Hermite polynomial

In [3,7,8,16], it is established that the umbral method serves as an important tool to deal with certain properties of special functions. In this paper, by making use of their method, we introduce the umbral deﬁnition of the 3-variable Hermite polynomials Hn(x, y, z). In this section, we also obtain the umbra for 3VHP Hn(x, y, z) and study some of its new properties.

Taking x = 0 and y = 0 in equation (13), we obtain the boundary condition for the 3-variable Hermite polynomials Hn(x, y, z):

Hn(0, 0, z) = zn3_n_!

(n₃+ 1)

2cosnπ₃ – |cosnπ|. (24) In view of equation (24), we introduce the following umbra:

ˆcr zψ0= z3rr! (r 3+ 1) 2cosrπ₃ – |cosrπ| (ψ0= 0), (25)

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It follows from Eq. (25) that

eˆczt_ψ 0= ezt

3

. (26)

Using equations (6) and (26) in equation (12), we get the following umbral form of the generating function of 3VHP Hn(x, y, z): ∞ n=0 Hn(x, y, z) tn n! = e (x+ˆby+ˆcz)t_φ 0ψ0, (27)

which on expanding the exponential function in the right-hand side and then comparing the equal powers of t from both sides of the resultant equation gives the following umbral deﬁnition of the 3-variable Hermite polynomials Hn(x, y, z):

Hn(x, y, z) = (x + ˆby+ˆcz)nφ0ψ0

= e(ˆcz+ˆby)Dx_xn_φ 0ψ0

= eˆczDx_H

n(x, y)ψ0,

where ˆbyis acting on φ0andˆczis acting on ψ0.

The use of the above equation allows a signiﬁcant simpliﬁcation of the theory of 3VHP

Hn(x, y, z), and it would be largely exploited in the ﬁeld of special functions. We note that such a point of view has opened new avenues in the derivation of lacunary generating functions and for the relevant combinatorial interpretation [12].

Now, we obtain the umbral deﬁnition and umbral operational deﬁnition of the 3-variable generalised Hermite polynomials (3VgHP) Hn(s,m)(x, y, z).

In view of equation (25), we introduce the following generalised form of umbraˆcz:

sˆcrzψ0= zrs_r_! (r_s+ 1)As,r, (28) where As,r= ⎧ ⎨ ⎩ 1 r= sp, p∈ N, 0, otherwise. (29)

If we take A3,r= (|2 cos rπ₃| – | cos rπ|), then for s = 3 equation (28) gives (25) andsˆczψ0

reduces toˆczψ0. By equation (28), we have esˆczt_ψ 0= ezt s . (30)

Using equations (23) and (30), we get

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which on further simpliﬁcation gives umbral operational deﬁnition of 3VgHP Hn(s,m)(x, y, z) as follows: H_n(s,m)(x, y, z) = e(sˆcz+mˆby)Dx_xn_φ 0ψ0 = exp(sˆczDx)(x +mˆby)nφ0ψ0, wheremˆby r

φ0andsˆczrψ0are deﬁned in equations (17) and (28), respectively.

By using Crofton identity given in [9]

eλDmx_f_{(x) = f}_x_{+ mλD}m–1 x

, (32)

we obtain 3VgHP Hn(s,m)(x, y, z) binomially as follows:

H_n(s,m)(x, y, z) = (x +mˆby+sˆcz)nφ0ψ0. (33)

In the next section, we generalise the variable Hermite polynomial to 4-parameters 3-variables Hermite polynomials arising from umbral method.

3 An extension of the 3-variable Hermite polynomials

It is realised that the advantage of umbral method is that this method serves as an impor-tant extension of certain special functions that cannot be extended by using classical op-erational method; see for example [14,15]. In this section, by using the fact that the power of these umbras can be any real numbers, we extend the 3-variable Hermite polynomials to 4-parameter 3-variable Hermite polynomials by using the Hermite umbras given as ˆby andˆczin equations (5) and (25), respectively.

Further, we study the properties of the 4-parameter 3-variable Hermite polynomials

Hn(x, y, z|β, α; p, q) and apply the umbral method to aforementioned polynomial. We introduce the 4-parameter 3-variable Hermite polynomials (4P3VHP)

Hn(x, y, z|β, α; p, q) given by Hn(x, y, z|β, α; p, q) = ˆbβyˆcpz

x+ ˆbα_y+ˆcq_znφ0ψ0, (34)

where α, β, p and q∈ N ∪ {0}.

By equation (34), we have the following generating function for 4P3VHP Hn(x, y, z|β, α; p, q).

Theorem 3.1 The generating function of4P3VHP Hn(x, y, z|β, α; p, q) is given by ∞ n=0 Hn(x, y, z|β, α; p, q) tn n! = e xt_yβ₂_e(α,β)_yα₂_t_zp₃_E p,q zq3_t_, ₍₃₅₎ where Ep,q(x) = ∞ r=0 (p + qr + 1)xr (p+qr₃ + 1)r! 2cos(p + qr)π₃ –cos(p + qr)π . (36)

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Proof From equation (34), we have ∞ n=0 tn n!Hn(x, y, z|β, α; p, q) = ∞ n=0 tn n!ˆb β yˆcpz x+ ˆbα_y+ˆcq_znφ0ψ0 = ˆbβ_yˆcp_ze(x+ˆbαy+ˆcqz)t_φ 0ψ0.

Since it is obvious that [x + ˆbα y,ˆc

q

z] = 0 and [x, ˆbαy] = 0 and using the Weyl decoupling identity [9]

eˆA+ˆB= eˆAeˆBe–k2_, _k_{= [ ˆ}_A_{, ˆB], (k}∈ C) ₍₃₇₎

in the above equation, we ﬁnd

∞ n=0 tn n!Hn(x, y, z|β, α; p, q) = ˆb β yˆcpzexteˆb α yt_eˆcqzt_φ 0ψ0, (38)

which, on expanding the exponentials in the right-hand side, gives

∞ n=0 tn n!Hn(x, y, z|β, α; p, q) = e xt ∞ r=0 ˆbαr+β y tr r! ∞ s=0 ˆcp+qs z ts s! φ0ψ0. Now, using equations (5) and (25), we get

∞ n=0 tn n!Hn(x, y, z|β, α; p, q) = extyβ2 ∞ r=0 (αr + β + 1)yαr2_tr (αr₂+β+ 1)r! cosαr+ β 2 π × zp3 ∞ s=0 (p + qs + 1)zqs3_ts (p+qs₃ + 1)s! 2cosπ(p + qs)₃ –cos(p + qs)π .

Using equations (10) and (36) in the right-hand side of the above equation, we get assertion

(35).

Remark3.1 The functionEp,q(x) is a generalisation of ex, as for p = 0 and q = 1 in equation

(36), we getE0,1(x) = ex3_.

Next, we obtain the following series deﬁnition for 4P3VHP Hn(x, y, z|β, α; p, q). Theorem 3.2 The series deﬁnition for4P3VHP Hn(x, y, z|β, α; p, q) is given by

Hn(x, y, z|β, α; p, q) = n! n r=0 (p + qr + 1)zp+qr3 _H_n_–r(x, y|β, α) (p+qr₃ + 1)r!(n – r)! ×2cos(p + qr)π 3 –cos(p + qr)π , (39)

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where Hn–r(x, y|β, α) denotes 2P2VHP given by means of the following generating function: ∞ n=0 Hn(x, y|β, α) tn n!= e xt_yβ₂_e(α,β)_yα₂_t_.

Proof From equation (38), we have

∞ n=0 tn n!Hn(x, y, z|β, α; p, q) = ˆb β yˆcpze (x+ˆbα y)t_eˆcqtz_φ 0ψ0,

which, on expanding exponentials in the right-hand side of the above equation, gives

∞ n=0 tn n!Hn(x, y, z|β, α; p, q) = ∞ n=0 ˆbβ y x+ ˆbα_ynt n n! ∞ r=0 ˆcp+qr z tr r!φ0ψ0 = ∞ n=0 n r=0 ˆbβ y(x + ˆbαy)n–rˆc p+qr z tn (n – r)!r! φ0ψ0.

Using equations (8) and (25), we get

∞ n=0 tn n!Hn(x, y, z|β, α; p, q) = ∞ n=0 n r=0 (p + qr + 1)zp+qr3 _H_n_–r(x, y|β, α) (p+qr₃ + 1)r!(n – r)! ×2cos(p + qr)π 3 –cos(p + qr)π tn.

Comparing the equal powers of t from both sides of the above equation, we get assertion

(39).

Further, we discuss an alternative formulation of the theory of the generalised Hermite polynomials using umbral formalism, which will be embedded with the technique devel-oped in this paper.

Now, we obtain the following result.

Theorem 3.3 The following formula for4P3VHP Hn(x, y, z|β, α; p, q) holds: Hn+k(x, y, z|β, α; p, q)) = k r=0 r s=0 k r r s xsHn x, y, z|α(r – s) + β, α; q(k – r) + p, q. (40)

Hn+k(x, y, z|β, α; p, q) = ˆbβyˆcpz x+ ˆbα y+ˆcqz n+k φ0ψ0 = ˆbβ_yˆcp_zx+ ˆbα_y+ˆcq_zkx+ ˆbα_y+ˆcq_znφ0ψ0.

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Expanding the ﬁrst bracket of the right-hand side of the above equation binomially, we have Hn+k(x, y, z|β, α; p, q)) = ˆbβyˆcpz k r=0 k r x+ ˆbα_yrˆcq_z(k–r)x+ ˆbα_y+ˆcq_znφ0ψ0.

Again, expanding the ﬁrst bracket of the right-hand side of the above equation binomially, we ﬁnd Hn+k(x, y, z|β, α; p, q)) = k r=0 r s=0 k r r s xsˆbα_y(r–s)+βˆcq_z(k–r)+px+ ˆbα_y+ˆcq_znφ0ψ0.

Using equation (34) in the right-hand side of the above equation, we get assertion

(40).

For k = n, Theorem3.3gives the following result.

Corollary 3.1 The following index duplication formula for4P3VHP

Hn(x, y, z|β, α; p, q) holds: H2n(x, y, z|β, α; p, q) = n r=0 r s=0 n r r s xsHn x, y, z|α(r – s) + β, α; q(n – r) + p, q. (41)

Further, we obtain the following result.

Theorem 3.4 The following argument duplication formula for4P3VHP Hn(x, y, z|β, α; p, q) holds: Hn(2x, y, z|β, α; p, q)) = n s=0 s r=0 r u=0 n s s r r u xu 1 2s–u × Hn–s x,y 2, z 2|α(r – u) + β, α; q(s – r) + p, q . (42)

Hn(2x, y, z|β, α; p, q) = ˆbβyˆcpz x+ ˆb α y 2 + ˆcq z 2 + x+ˆb α y 2 + ˆcq z 2 n φ0ψ0,

which on simpliﬁcation gives

Hn(2x, y, z|β, α; p, q) = ˆbβyˆcpz n s=0 n s x+ˆb α y 2 + ˆcq z 2 n–s x+ ˆb α y 2 + ˆcq z 2 s φ0ψ0.

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Expanding the second bracket in right-hand side of the above equation binomially, we ﬁnd Hn(2x, y, z|β, α; p, q) = ˆbβ yˆcpz n s=0 s r=0 n s s r x+ ˆb α y 2 + ˆcq z 2 n–s x+ˆb α y 2 r_ˆcq z 2 s–r φ0ψ0,

which on further simpliﬁcation gives

Hn(2x, y, z|β, α; p, q) = ˆbβyˆcpz n s=0 s r=0 r u=0 n s s r r u xu ˆb α y 2 r–u_ˆcq z 2 s–r × x+ˆb α y 2 + ˆcq z 2 n–s φ0ψ0.

Using equation (34) in the right-hand side of the above equation, we get assertion (42). Now, we ﬁnd the following series representation of the 4-parameter 2-variable Hermite polynomials in terms of the 4-parameter 3-variable Hermite polynomials.

Theorem 3.5 The series deﬁnition of4P2VHP Hn(x, y|β, α; p, q) is given by

Hn(x, y|β, α; p, q) = n r=0 n r (–1)rHn–r(x, y, z|β, α; p + qr, q). (43)

Hn(x, y|β, α; p, q) = ˆbβyˆcpz x+ ˆbα y+ˆcqz–ˆcqz n φ0ψ0, (44)

from which, on expanding binomially, we get

Hn(x, y|β, α; p, q) = ˆbβyˆcpz n r=0 n r (–1)rx+ ˆbα_y+ˆcq_zn–rˆcqr_z φ0ψ0. (45)

Using equation (34) in the above equation we get assertion (43).

Remark3.2 For taking p = 0 and q = 1 in equation (44) of Theorem3.5, we get the follow-ing series representation of 2P2VHP Hn(x, y,|β, α):

Hn(x, y|β, α) = n r=0 n r (–1)r_H n–r(x, y, z|β, α; r, 1). (46)

Remark3.3 For taking β = 0, α = 1, p = 0 and q = 1 in equation (44) of Theorem3.5, we get the following series representation of 2VHP Hn(x, y):

Hn(x, y) = n r=0 n r (–1)rHn–r(x, y, z|–, 1; r, 1). (47)

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Now, we obtain the operational deﬁnition of 4P3VHP Hn(x, y, z|β, α; p, q).

Since DxHn(x, y) = nHn–1(x, y) and DxHn(x, y, z) = nHn–1(x, y, z), it can be veriﬁed that Hn(x, y, z|β, α; p, q) = y β 2_e (α,β) yα2_D x zp3E p,q zq3_D x xn, and for taking α = 1 and β = 0, we have

Hn(x, y, z|–, 1; p, q) = z p 3E p,q zq3_D x Hn(x, y).

In the next section, we consider some special cases of the results established in this section. 4 Special cases

In this section, we obtain certain new as well as known special polynomials by using suit-able choices for parameters and varisuit-able z in equations (34), (35) and (39) as special cases of 4-parameter 3-variable Hermite polynomials.

In the following table, the umbral deﬁnitions, generating functions and series deﬁnitions of certain polynomials are listed.

For the same choices of parameters α, β, p and q considered in Table1, equations (41) and (42) give the index duplication and argument duplication formulas for the special polynomials mentioned in the same table. The respective formulas are listed in Table2.

In the concluding remarks, we present further argument supporting the eﬀectiveness of the umbral method.

Table 1 Some new and known special polynomials

S. No. Para-meters

Polynomials Umbral deﬁnition Generating function Series deﬁnition

I. q = 1 Hn(x, y, z|β,α; p, 1) ˆbβ_{y ˆc}p_{z (x + ˆb}α_{y + ˆcz )}nφ0ψ0 ext y β 2 e(α,β)(y α 2 t)z p 3_Ep,1(z13 t) n!n_r=0(p+r+1)z p+r 3 Hn–r(x,y|β,α) (p+r₃ +1)r!(n–r)! × (|2 cos(p+r)π 3 | – | cos(p + r)π|)

II. α= 1 Hn(x, y, z|β, 1; p, q) ˆbβ_{y ˆc}p_{z (x + ˆby + ˆc}q_{z )}nφ0ψ0 ext y

β 2 e(1,β)(y 1 2 t)z p 3_Ep,q(z q 3 t) n!n_r=0(p+qr+1)z p+qr 3 _{Hn–r (x,y|}β,1) (p+qr₃ +1)r!(n–r)! × (|2 cos(p+qr)3 π| – | cos(p + qr)π|) III. α= 1; q = 1 Hn(x, y, z|β, 1; p, 1) ˆbβ_{y ˆc}p_{z (x + ˆby + ˆcz )}nφ0ψ0 ext y β 2 e(1,β)(y 1 2 t)z p 3_Ep,1(z13 t) n!n_r=0(p+r+1)z p+r 3 Hn–r(x,y|β,1) (p+r₃ +1)r!(n–r)! × (|2 cos(p+r)3π| – | cos(p + r)π|) IV. p = 0 Hn(x, y, z|β,α; –, q) ˆbβ_{y (x + ˆb}α_{y + ˆc}q_{z )}nφ0ψ0 ext y β 2 e(α,β)(y α 2 t)_E0,q(z q 3 t) n!n_r=0(qr+1)z qr 3 Hn–r(x,y|β,α) (qr₃+1)r!(n–r)! × (|2 cos(qr)π 3 | – | cos(qr)π|) V. p = 0; q = 1 Hn(x, y, z|β,α; –, 1) ˆbβ_{y (x + ˆb}α_{y + ˆcz )}nφ0ψ0 ext y β 2 e(α,β)(y α 2 t)ezt3 n! n3_r=0zr Hn–3r(x,y|_r!(n–3r)!β,α) VI. α= 1; p = 0 Hn(x, y, z|β, 1; –, q) ˆbβy (x + ˆby + ˆcqz )nφ0ψ0 ext y β 2 e(1,β)(y 1 2 t)_E0,q(zq3 t) n!n_r=0(qr+1)z qr 3 Hn–r(x,y|β,1) (qr₃+1)r!(n–r)! × (|2 cos(qr)π 3 | – | cos(qr)π|) VII. α= 1; p = 0; q = 1 Hn(x, y, z|β, 1; –, 1) ˆbβ_{y (x + ˆby + ˆcz )}nφ0ψ0 ext y β 2 e(1,β)(y 1 2 t)ezt3 n![ n3 ]_r=0zr Hn–3r(x,y|β;1) r!(n–3r)!

VIII. β= 0 Hn(x, y, z|–,α; p, q) ˆcp_{z (x + ˆb}_{y + ˆc}α q_{z )}nφ0ψ0 ext e(α,0)(y

α 2 t)z p 3_Ep,q(z q 3 t) n!n_r=0(p+qr+1)z p+qr 3 _{Hn–r (x,y|–,}α) (p+qr₃ +1)r!(n–r)! × (|2 cos(p+qr)π 3 | – | cos(p + qr)π|)

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Table 1 (Continued)

S. No. Para-meters

Polynomials Umbral deﬁnition Generating function Series deﬁnition

IX. β= 0; q = 1 Hn(x, y, z|–,α; p, 1) ˆcpz (x + ˆbαy + ˆcz )nφ0ψ0 ext e(α,0)(y α 2 t)z p 3_Ep,1(z13 t) n!n_r=0(p+r+1)z p+r 3 Hn–r(x,y|–,α) (p+r₃ +1)r!(n–r)! × (|2 cos(p+r)3π| – | cos(p + r)π|) X. α= 1; β= 0 Hn(x, y, z|–, 1; p, q) ˆcpz (x + ˆby + ˆcqz )nφ0ψ0 ext+yt 2 z p 3_Ep,q(z q 3 t) n!n r=0 (p+qr+1)z p+qr 3 _{Hn–r (x,y)} (p+qr₃ +1)r!(n–r)! × (|2 cos(p+qr)3 π| – | cos(p + qr)π|) XI. α= 1; β= 0; q = 1 Hn(x, y, z|–, 1; p, 1) ˆcpz (x + ˆby + ˆcz )nφ0ψ0 ext+yt2z p 3_Ep,1(z13 t) n!n_r=0(p+r+1)z p+r 3 Hn–r(x,y) (p+r 3 +1)r!(n–r)! × (|2 cos(p+r)3π| – | cos(p + r)π|) XII. β= 0; p = 0 Hn(x, y, z|–,α; –, q) (x + ˆbα_{y + ˆc}q_{z )}nφ0ψ0 ext e(α,0)(y α 2 t)_E0,q(z q 3 t) n!n_r=0(qr+1)z qr 3 Hn–r(x,y|–,α) (qr₃+1)r!(n–r)! × (|2 cos(qr)π 3 | – | cos(qr)π|) XIII. β= 0; p = 0; q = 1 Hn(x, y, z|–,α; –, 1) (x + ˆbα_{y + ˆcz )}nφ0ψ0 ext e(α,0)(y α 2 t)ezt3 n![ n3 ]_r=0zr Hn–3r(x,y|–,α) r!(n–3r)! XIV. α= 1; β= 0; p = 0 Hn(x, y, z|–, 1; –, q) (x + ˆby + ˆcqz )nφ0ψ0 ext+yt 2 E0,q(zq3 t) n!n_r=0(qr+1)z qr 3 Hn–r(x,y) (qr₃+1)r!(n–r)! × (|2 cos(qr)π 3 | – | cos(qr)π|) XV. α= 1; β= 0; p = 0; q = 1

Hn(x, y, z) (x + ˆby + ˆcz )nφ0ψ0 e(xt+yt2+zt3)[6] n![ n3 ]_r=0zr Hn–3r(x,y) r!(n–3r)! [6] XVI. p = 0; q = 0; z = 0 Hn(x, y|β,α) ˆbβ_{y (x + ˆb}α_{y )}nφ0[8] ext y β 2 e(α,β)(y α 2 t)[8] n!n_r=0(αr+β+1)xn–r y αr+β 2 (αr+β 2 +1)r!(n–r)! × |(cosαr+β 2 π)| XVII. p = 0; q = 0; α= 1; z = 0 Hn(x, y|β, 1) ˆbβ_{y (x + ˆby )}nφ0 ext y β 2 e(1,β)(y 1 2 t) n!n_r=0(r+β+1)xn–r y r+β 2 (r+₂β+1)r!(n–r)! × |(cosr+β 2 π)| XVIII. β= 0; p = 0; q = 0; z = 0

Hn(x, y|–,α) (x + ˆbα_{y )}nφ0 ext e(α,0)(y

α 2 t) n!n_r=0(αr+1)xn–r y αr 2 (α₂r+1)r!(n–r)! × |(cos αr 2π)| XIX. β= 0; p = 0; α= 1; q = 0; z = 0 Hn(x, y) (x + ˆby )nφ0[8] ext+yt 2 [2] n! n2_r=0xn–2r yr r!(n–2r)![2] 5 Concluding remarks

Gaussian integral representation of Hermite polynomials as well as speciﬁc umbral meth-ods play an important role in classical problems arising in quantum optics, quantum me-chanics, biomathematics and engineering (see for example [1,17–20]). They are exploited to calculate the optical mode overlapping and transition rates between quantum eigen-states of the harmonic oscillator. A general method allowing the direct evaluation of these integrals has not been developed. Babusci et al. described a unifying method, ﬂexible for generalisation, which provides a direct method for the evaluation of this class of integrals [4,5].

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Table 2 Index and argument duplication formulas

S. No. Polynomials Index duplication formula Argument duplication formula I. Hn(x, y, z|β,α; p, 1) nr=0 r s=0 n r r s xs_H n(x, y, z|α(r – s) +β,α; p + n – r, 1) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,2y, z 2|α(r – u) +β,α; s – r + p, 1) II. Hn(x, y, z|β, 1; p, q) nr=0 r s=0 n r r s xs_H n(x, y, z|r – s +β, 1; q(n – r) + p, q) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,2y, z 2|r – u +β, 1; q(s – r) + p, q) III. Hn(x, y, z|β, 1; p, 1) nr=0 r s=0 n r r s xs_H n(x, y, z|r – s +β, 1; n – r + p, 1) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,2y, z 2|r – u +β, 1; s – r + p, 1) IV. Hn(x, y, z|β,α; –, q) nr=0 r s=0 n r r s xs_H n(x, y, z|α(r – s) +β,α; q(n – r), q) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,2y, z 2|α(r – u) +β,α; q(s – r), q) V. Hn(x, y, z|β,α; –, 1) nr=0 r s=0 n r r s xs_H n(x, y, z|α(r – s) +β,α; n – r, 1) ns=0 s r=0 r u=0 n s s r r u xs–r–u 1 2s–u× Hn–s(x,2y, z 2|α(r – u) +β,α; s – r, 1) VI. Hn(x, y, z|β, 1; –, q) nr=0 r s=0 n r r s xs_H n(x, y, z|r – s +β, 1; q(n – r), q) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,2y, z 2|r – u +β, 1; q(s – r), q) VII Hn(x, y, z|β, 1; –, 1) nr=0 r s=0 n r r s xs_H n(x, y, z|r – s +β, 1; n – r, 1) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|r – u +β, 1; s – r, 1) VIII. Hn(x, y, z|–,α; p, q) nr=0 r s=0 n r r s xs_H n(x, y, z|α(r – s),α; q(n – r) + p, q) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|α(r – u),α; q(s – r) + p, q) IX. Hn(x, y, z|–,α; p, 1) nr=0 r s=0 n r r s xs_H n(x, y, z|α(r – s),α; n – r + p, 1) ns=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|α(r – u),α; s – r + p, 1) X. Hn(x, y, z|–, 1; p, q) n r=0 r s=0 n r r s xs_H n(x, y, z|r – s, 1; q(n – r) + p, q) n s=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|r – u, 1; q(s – r) + p, q) XI. Hn(x, y, z|–, 1; p, 1) n r=0 r s=0 n r r s xs_H n(x, y, z|r – s, 1; n – r + p, 1) n s=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|r – u, 1; s – r + p, 1) XII. Hn(x, y, z|–,α; –, q) n r=0 r s=0 n r r s xs_H n(x, y, z|α(r – s),α; q(n – r), q) n s=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|α(r – u),α; q(s – r), q) XIII. Hn(x, y, z|–,α; –, 1) n r=0 r s=0 n r r s xs_H n(x, y, z|α(r – s),α; n – r, 1) n s=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|α(r – u),α; s – r, 1) XIV. Hn(x, y, z|–, 1; –, q) n r=0 r s=0 n r r s xs_H n(x, y, z|r – s, 1; q(n – r), q) n s=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|r – u, 1; q(s – r), q) XV. Hn(x, y, z) n r=0 r s=0 n r r s xs_H n(x, y, z|r – s, 1; n – r, 1) n s=0 s r=0 r u=0 n s s r r u xu 1 2s–u× Hn–s(x,₂y,z₂|r – u, 1; s – r, 1) XVI. Hn(x, y|β,α) n r=0 n r xr_H n(x, y|α(n – r) +β,α) n s=0 s r=0 n s s r xr 1 2s–r× Hn–s(x,₂y|α(s – r) +β,α) XVII. Hn(x, y|β, 1) n r=0 n r xr_H n(x, y|n – r +β, 1) n s=0 s r=0 n s s r xr 1 2s–r× Hn–s(x,y2|s – r +β, 1) XVIII. Hn(x, y|–,α) n r=0 n r xr_H n(x, y|α(n – r),α) n s=0 s r=0 n s s r xr 1 2s–r× Hn–s(x,₂y|α(s – r),α) XIX. Hn(x, y) n r=0 n r xn–r_H n(x, y|r) [8] n s=0 s r=0 n s s r xr 1 2s–r×Hn–s(x,y2|s–r) [8]

We consider the following integral:

In= ∞ –∞Hn(ax + b, y, z|β, α; p, q)e –cx2+ξ x_dx_. ₍₄₈₎ By equation (48), we have ∞ n=0 In tn n!= ∞ n=0 ∞ –∞Hn(ax + b, y, z|β, α; p, q) tn n!e –cx2_{+ξ x} dx,

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which on using equation (35) gives ∞ n=0 In tn n!= e bt_yβ₂_e(α,β)_yα₂_t_zp₃_E p,q zq3_t ∞ –∞e (at+ξ )x–cx2_dx_. ₍₄₉₎ Since ∞ –∞e bx–ax2_+c dx= √ π √ ae b2 4a+c_,

see [5], presenting the Gaussian integral, we ﬁnd

∞ n=0 In tn n!= e bt_yβ₂_e(α,β)_yα₂_t_zp₃_E p,q zq3_t√π√ c exp a2 4ct 2₊ξ2 4c+ aξ 2ct . (50)

Using equations (1) and (35) in the right-hand side of the above equation, we obtain

∞ n=0 In tn n!= √ π √ c exp ξ2 4c ∞ n=0 ∞ r=0 Hn(b, y, z|β, α; p, q)Hr aξ 2c, a2 4c tn+r n!r!.

Next, comparing the equal powers of t from both sides of the above equation, we get

In= √ π √ cexp ξ2 4c n r=0 n r Hn–r(b, y, z|β, α; p, q)Hr aξ 2c, a2 4c . (51)

In view of equations (48) and (51), we get the following result: ∞ –∞Hn(ax + b, y, z|β, α; p, q)e –cx2+ξ x_dx = √ π √ c exp ξ2 4c × n r=0 n r Hn–r(b, y, z|β, α; p, q)Hr aξ 2c, a2 4c .

Again, using equation (35) in the right-hand side of equation (50), we ﬁnd

∞ n=0 In tn n!= √ π √ c exp ξ2 4c ∞ n=0 ∞ r=0 Hn b+aξ 2c, y, z|β, α; p, q a2r (4c)r tn+2r n!r!. Comparing the equal powers of t from both sides of the above equation, we get

In= √ π √ cexp ξ2 4c n! [n₂] r=0 1 (n – 2r)!r!Hn–2r b+aξ 2c, y, z|β, α; p, q a2r (4c)r. (52) In view of equations (48) and (52), we get the following result:

∞ –∞Hn(ax + b, y, z|β, α; p, q)e –cx2+ξ x_dx = √ π √ c exp ξ2 4c n!× [n₂] r=0 1 (n – 2r)!r!Hn–2r b+aξ 2c, y, z|β, α; p, q a2r (4c)r.

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Similarly, for the same choices of parameters α, β, p and q considered in Table1, we can evaluate the integrals involving the special polynomials mentioned in the same table.

Acknowledgements

We would like to thank the reviewers for their valuable suggestions and comments. Funding

None.

Availability of data and materials Not applicable.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript. Author details

1_{Mathematics Section, Women’s College, Aligarh Muslim University, 202002, Aligarh, India.}2_{Department of Mathematics,}

Aligarh Muslim University, 202002, Aligarh, India. 3_{Department of Economics, Faculty of Economics, Administrative and}

Social Sciences, Hasan Kalyoncu University, TR-27410, Gaziantep, Turkey. 4_{Department of Basic Engineering Sciences,}

Engineering Faculty, Malatya Turgut Ozal University, 44040, Malatya, Turkey.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 9 October 2020 Accepted: 4 November 2020

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