International Journal of Analysis and Applications

Volume 16, Number 6 (2018), 856-867

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-856

IMPLICIT SUMMATION FORMULA FOR 2-VARIABLE LAGUERRE-BASED

POLY-GENOCCHI POLYNOMIALS

WASEEM A. KHAN∗, IDREES A. KHAN AND MOIN AHMAD

Department of Mathematics, Faculty of Science, Integral University, Lucknow-226026, India

∗Corresponding author: waseem08 khan@rediffmail.com

Abstract. The main object of this paper is to introduce a new class of Laguerre-based poly-Genocchi

polynomials and investigate some properties for these polynomials and related to the Stirling numbers of

the second kind. We derive summation formulae and general symmetry identities by using different analytical

means and applying generating functions.

1. Introduction

The generalized Bernoulli, Euler and Genocchi polynomials of (real or complex) order α are usually defined

by means of the following generating functions (see [1-16]):(
t

et − 1

)α
ext =

∞∑
n=0

B(α)n (x)
tn

n!
, (| t |< 2π; 1α = 1), (1.1)

(
2

et + 1

)α
ext =

∞∑
n=0

E(α)n (x)
tn

n!
, (| t |< π; 1α = 1) (1.2)

and (
2t

et + 1

)α
ext =

∞∑
n=0

G(α)n (x)
tn

n!
, (| t |< π; 1α = 1). (1.3)

Received 2017-09-19; accepted 2017-12-07; published 2018-11-02.

2010 Mathematics Subject Classification. 33C45, 33C99, 05A10, 05A15.

Key words and phrases. Laguerre polynomials, poly-Genocchi polynomials, Laguerre-based poly-Genocchi polynomials,

summation formulae, symmetric identities.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

856

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-856


Int. J. Anal. Appl. 16 (6) (2018) 857

So that obviously

Bn(x) = B
1
n(x),En(x) = E

1
n(x) and Gn(x) = G

1
n(x), (n ∈ N),

where N0 = N∪{0}(N = 1, 2, 3, · · ·).

The classical polylogarithmic function Lik(z) is defined by (see [2], [10]):

Lik(z) =

∞∑
m=1

zm

mk
, (k ∈ Z). (1.4)

The poly-Bernoulli numbers and polynomials are defined by following generating functions (see [7], [8],

[9]):

Lik(1 −e−t)
et − 1

=
∞∑
n=0

B(k)n
tn

n!
, (1.5)

Lik(1 −e−t)
et − 1

ext =

∞∑
n=0

B(k)n (x)
tn

n!
. (1.6)

In the case k = 1 in (1.5) and (1.6), we have

B(1)n = Bn,B
(1)
n (x) = Bn(x).

The poly-Genocchi numbers and polynomials are defined by following generating functions (see [14]):

2Lik(1 −e−t)
et + 1

=

∞∑
n=0

G(k)n
tn

n!
, (1.7)

2Lik(1 −e−t)
et + 1

ext =

∞∑
n=0

G(k)n (x)
tn

n!
. (1.8)

In the case k = 1 in (1.7) and (1.8), we have

G(1)n = Gn,G
(1)
n (x) = Gn(x).

The 2-variable Laguerre polynomials (2-VLP)  Ln(x,y), which is defined by (see [5]):

1

(1 −yt)
exp

(
−xt

1 −yt

)
=

∞∑
n=0

 Ln(x,y)t
n, (| yt |< 1) (1.9)

It is equivalently given by (see [6]).

exp(yt)C0(xt) =

∞∑
n=0

 Ln(x,y)
tn

n!
, (1.10)

where C0(x) denotes the 0
th order Tricomi function. The nth order Tricomi functions Cn(x) are defined as:

Cn(x) =

∞∑
r=0

(−1)rxr

r!(n + r)!
, (n ∈ N0) (1.11)



Int. J. Anal. Appl. 16 (6) (2018) 858

with the following generating function:

exp
(
t−

x

t

)
=

∞∑
n=0

Cn(x)t
n, (1.12)

for t 6= 0 and for all finite x.

From (1.9) and (1.10), we get

Ln(x,y) = n!

n∑
s=0

(−1)sxsyn−s

(s!)2(n−s)!
= ynLn(x/y). (1.13)

Thus, we have

 Ln(x, 0) =
(−1)nxn

n!
,  Ln(0,y) = y

n,  Ln(x, 1) =  Ln(x), (1.14)

where Ln(x) are the classical Laguerre polynomials (see [1]).

Now, we recall here the following definition as follows:

The Stirling number of the first kind is given by

(x)n = x(x− 1) · · ·(x−n + 1) =
n∑
l=0

S1(n,l)x
l, (n ≥ 0) (1.15)

and the Stirling number of the second kind is defined by generating function:

(et − 1)n = n!
∞∑
l=n

S2(l,n)
tl

l!
. (1.16)

2. 2-Variable Laguerre-based poly-Genocchi polynomials

Let k ∈ Z, we inroduce 2-variable Laguerre-based poly-Genocchi polynomials by the following generating

function:

2Lik(1 −e−t)
et + 1

exp(yt)C0(xt) =

∞∑
n=0

LG
(k)
n (x,y)

tn

n!
, (2.1)

so that

LG
(k)
n (x,y) =

n∑
m=0


 n

m


G(k)n−mLm(x,y). (2.2)

When x = y = 0, LG
(k)
n = G

(k)
n (0, 0) are called the poly-Genocchi numbers. For k = 1 in (2.1), we have

2Li1(1 −e−t)
et + 1

exp(yt)C0(xt) =

∞∑
n=0

LGn(x,y)
tn

n!
, (2.3)

where LGn(x,y) is Laguerre-based Genocchi polynomials (see [13]).

Thus, we have

LG
(k)
n (x,y) = LGn(x,y), (n ≥ 0).



Int. J. Anal. Appl. 16 (6) (2018) 859

On setting x = 0, (2.1) reduces to the known result of Kim et al. [14.,p.Eq.(4)4776]:

2Lik(1 −e−t)
et + 1

exp(yt) =

∞∑
n=0

G(k)n (y)
tn

n!
, (k ∈ Z). (2.4)

Theorem 2.1. The following explicit summation formulae for Laguerre-based poly-Genocchi polynomials

holds true:

LG
(2)
n (x,y) =

n∑
m=0


 n

m


 Bmm!
m + 1

LGn−m(x,y). (2.5)

Proof. Using generating function for Laguerre-based poly-Genocchi polynomials (2.1), we have

∞∑
n=0

LG
(k)
n (x,y)

tn

n!
=

2Lik(1 −e−t)
et + 1

exp(yt)C0(xt)

=
2

et + 1
exp(yt)C0(xt)

∫ t
0

1

ez − 1

∫ t
0

1

ez − 1
· · ·

1

ez − 1

∫ t
0

z

ez − 1
dz · · ·dz.

In particular k = 2, we have

∞∑
n=0

LG
(2)
n (x,y)

tn

n!
=

2

et + 1
exp(yt)C0(xt)

∫ t
0

z

ez − 1
dz =

(
∞∑
m=0

tmBm
m + 1

)
2t

et + 1
exp(yt)C0(xt)

=

(
∞∑
m=0

tmBmm!

m + 1m!

)(
∞∑
n=0

LGn(x,y)
tn

n!

)
.

Replacing n by n−m in the r.h.s of above equation, we have

∞∑
n=0

LG
(2)
n (x,y)

tn

n!
=

∞∑
n=0


 n∑
m=0


 n

m


 Bmm!
m + 1

LGn−m(x,y)


 tn
n!
.

On equating the coefficients of the like powers of t in both sides, we get (2.5).

Remark 2.1. On setting x = 0, Theorem (2.1) reduces to the known result of Kim et al. [14.,p. 4777,

Theorem (2.1)].

Corollary 2.1. For n ≥ 0, we have

G(2)n (y) =

n∑
m=0


 n

m


 Bmm!
m + 1

Gn−m(y). (2.6)

Theorem 2.2. For n ≥ 1, the degree of LG
(k)
n (x,y) is n-1. We have

LG
(k)
n (x,y)

n
=

n−1∑
m=0


 n− 1

m


 G(k)m+1
m + 1

Ln−m−1(x,y). (2.7)

Proof. From (2.1), we have

∞∑
n=0

LG
(k)
n (x,y)

tn

n!
=

2Lik(1 −e−t)
1 −e−t

exp(yt)C0(xt)



Int. J. Anal. Appl. 16 (6) (2018) 860

=

(
∞∑
m=0

G(k)m
tm

m!

)(
∞∑
n=0

Ln(x,y)
tn

n!

)
Replacing n by n−m in above equation and comparing the coefficients of tn, we get

LG
(k)
n (x,y) =

n∑
m=0


 n

m


G(k)m Ln−m(x,y), (n ≥ 0). (2.8)

From (2.8), we have

LG
(k)
n (x,y)

n
=

n−1∑
m=0


 n− 1

m


 G(k)m+1
m + 1

Ln−m−1(x,y), (n ≥ 1) (2.9)

Therefore by (2.9), we obtain the result (2.7).

Remark 2.2. For x = 0, Theorem (2.2) reduces to the known result of Kim et al. [14.,p. 4778, Theorem

(2.2)].

Corollary 2.2. For n ≥ 1, the degree of G(k)n (x) is n-1. We have

G
(k)
n (y)

n
=

n−1∑
m=0


 n− 1

m


 G(k)m+1
m + 1

yn−m−1. (2.10)

Theorem 2.3. For n ≥ 0, we have

LG
(k)
n (x,y) =

n∑
p=0

p+1∑
l=1

(−1)l+p+1l!S2(p + 1, l)
lk(p + 1)


 n

p



LGn−p(x,y). (2.11)

Proof. By using (2.1), we can be written as

∞∑
n=0

LG
(k)
n (x,y)

tn

n!
=

(
Lik(1 −e−t)

t

)(
2t

et + 1
exp(yt)C0(xt)

)
. (2.12)

Now

1

t
Lik(1 −e−t) =

1

t

∞∑
l=1

(1 −e−t)l

lk
=

1

t

∞∑
l=1

(−1)l

lk
(1 −e−t)l

=
1

t

∞∑
l=1

(−1)l

lk
l!

∞∑
p=l

(−1)pS2(p,l)
tp

p!

=
1

t

∞∑
p=1

p∑
l=1

(−1)l+p

lk
l!S2(p,l)

tp

p!

=

∞∑
p=0

(
p+1∑
l=1

(−1)l+p+1

lk
l!
S2(p + 1, l)

p + 1

)
tp

p!
. (2.13)

From equations (2.12) and (2.13), we get

∞∑
n=0

LG
(k)
n (x,y)

tn

n!
=

∞∑
p=0

(
p+1∑
l=1

(−1)l+p+1

lk
l!
S2(p + 1, l)

p + 1

tp

p!

)(
∞∑
n=0

LGn(x,y)
tn

n!

)
.



Int. J. Anal. Appl. 16 (6) (2018) 861

Replacing n by n−p in the r.h.s. of above equation and comparing the coefficients of tn in both sides,

we arrive at the desired result (2.11).

Remark 2.3. For x = 0, Theorem (2.3) reduces to the known result of Kim et al. [14.,p. 4779, Theorem

(2.3)].

Corollary 2.3. For n ≥ 0, we have

G(k)n (y) =

n∑
p=0

p+1∑
l=1

(−1)l+p+1l!S2(p + 1, l)
lk(p + 1)


 n

p


Gn−p(y). (2.14)

Theorem 2.4. For n ≥ 1, we have

LG
(k)
n (x,y + 1) + LG

(k)
n (x,y) = 2

n∑
p=1

p∑
l=1

(−1)l+p

lk
l!S2(p,l)


 n

p


Ln−p(x,y). (2.15)

Proof. By using definition (2.1), we have

∞∑
n=0

LG
(k)
n (x,y + 1)

tn

n!
+

∞∑
n=0

LG
(k)
n (x,y)

tn

n!

=
2Lik(1 −e−t)

et + 1
exp((y + 1)t)C0(xt) +

2Lik(1 −e−t)
et + 1

exp(yt)C0(xt)

= 2Lik(1 −e−t) exp(y)tC0(xt)

=

∞∑
p=1

(
2

p∑
l=1

(−1)l+p

lk
l!S2(p,l)

)
tp

p!
exp(yt)C0(xt)

=

(
∞∑
p=1

(
2

p∑
l=1

(−1)l+p

lk
l!S2(p,l)

)
tp

p!

)(
∞∑
n=0

Ln(x,y)
tn

n!

)
.

Replacing n by n−p in the above equation and comparing the coefficients of tn in both sides, we obtain

the result (2.15).

Remark 2.4. Taking x = 0, Theorem 2.4 reduces to the known result of Kim et al. [14.,p. 4780, Theorem

(2.4)].

Corollary 2.4. For n ≥ 1, we have

G(k)n (y + 1) + G
(k)
n (y) = 2

n∑
p=1

p∑
l=1

(−1)l+p

lk
l!S2(p,l)


 n

p


yn−p. (2.16)

Theorem 2.5. For d ∈ N with d ≡ 1(mod2), we have

LG
(k)
n (x,y) =

n∑
p=0


 n

p


dn−p−1 p+1∑

l=0

d−1∑
a=0

(−1)l+p+1l!S2(p + 1, l)
lk

(−1)aLGn−p(
a + y

d
,x). (2.17)



Int. J. Anal. Appl. 16 (6) (2018) 862

Proof. From equation (2.1), we can be written as

∞∑
n=0

LG
(k)
n (x,y)

tn

n!
=

2Lik(1 −e−t)
et + 1

exp(yt)C0(xt)

=

(
2Lik(1 −e−t)

t

)(
2t

edt + 1

d−1∑
a=0

(−1)a exp((a + y)t)C0(xt)

)

=

(
∞∑
p=0

(
p+1∑
l=1

(−1)l+p+1

lk
l!
S2(p + 1, l)

p + 1

)
tp

p!

)(
∞∑
n=0

dn−1
d−1∑
a=0

(−1)aLGn(
a + y

d
,x)

tn

n!

)
.

Replacing n by n−p in above equation and comparing the coefficient of tn in both sides, we get (2.17).

Remark 2.5. For x = 0, Theorem 2.5 reduces to the known result of Kim et al. [14.,p. 4780].

Corollary 5. For d ∈ N with d ≡ 1(mod2), we have

G(k)n (y) =

n∑
p=0


 n

p


dn−p−1 p+1∑

l=0

d−1∑
a=0

(−1)l+p+1l!S2(p + 1, l)
lk

(−1)aGn−p(
a + y

d
). (2.18)

3. Summation formulae for Laguerre-based poly-Genocchi polynomials

In this section, we establish summation formula for Laguerre-based poly-Genocchi polynomials by using

series techniques method.

Theorem 3.1. The following implicit summation formulae for Laguerre-based poly-Genocchi polynomials

LG
(k)
n (x,y) holds true:

LG
(k)
l+p(x,z) =

l,p∑
m,n=0


 l

m




 p

n


 (z −y)m+nLG(k)l+p−m−n(x,y). (3.1)

Proof. Replacing t by t + u and rewrite the generating function (2.1) as

2Lik(1 − (e)−(t+u))
et+u + 1

C0(x(t + u)) = e
−y(t+u)

∞∑
l,p=0

LG
(k)
l+p(x,y)

tl

l!

up

p!
. (3.2)

Replacing y by z in the above equation and equating the resulting equation to the above equation, we get

e(z−y)(t+u)
∞∑

m,l=0

LG
(k)
l+p(x,y)

tl

l!

up

p!
=

∞∑
l,p=0

LG
(k)
l+p(x,z)

tl

l!

up

p!
. (3.3)

On expanding exponential function (3.3) gives

∞∑
N=0

[(z −y)(t + u)]N

N!

∞∑
l,p=0

LG
(k)
l+p(x,y)

tl

l!

up

p!
=

∞∑
l,p=0

LG
(k)
l+p(x,z)

tl

l!

up

p!
(3.4)

which on using formula [16, p.52(2)]

∞∑
N=0

f(N)
(x + y)N

N!
=

∞∑
n,m=0

f(n + m)
xn

n!

ym

m!
, (3.5)



Int. J. Anal. Appl. 16 (6) (2018) 863

in the left hand side becomes

∞∑
m,n=0

(z −x)m+ntmun

m!n!

∞∑
l,p=0

HG
(k)
l+p(x,y)

tl

l!

up

p!
=

∞∑
l,p=0

HG
(k)
l+p(z,y)

tl

l!

up

p!
(3.6)

Now replacing l by l−m, p by p−n and using the lemma [16, p.100(1)] in the left hand side of (3.6), we get

∞∑
m,n=0

∞∑
l,p=0

(z −x)m+n

m!n!
LG

(k)
l+p−m−n(x,y)

tl

(l−m)!
up

(p−n)!

=

∞∑
l,p=0

LG
(k)
l+p(x,z)

tl

l!

up

p!
. (3.7)

Finally on equating the coefficients of the like powers of t and u in the above equation, we get the required

result.

Remark 3.1. Taking l = 0 in assertion (3.1) of Theorem 3.1, we deduce the following consequence of

Theorem 3.1.

Corollary 3.1. The following summation formula for Laguerre-based poly-Genocchi polynomials HG
(k)
n (z,y)

holds true:

LG
(k)
p (x,z) =

p∑
n=0


 p

n


 (z −y)nLG(k)p−n(x,y). (3.8)

Remark 3.2. Replacing z by z + y in (3.8), we obtain

LG
(k)
p (x,z + y) =

p∑
n=0


 p

n


znLG(k)p−n(x,y). (3.9)

Theorem 3.2. The following summation formula for Laguerre-based poly-Genocchi polynomials HG
(k)
n (z,y)

holds true:

LG
(k)
n (x,y + u) =

n∑
j=0


 n

j


ujLG(k)n−j(x,y). (3.10)

Proof. Using (2.1), we can be written as

∞∑
n=0

LG
(k)
n (x,y + u)

tn

n!
=

2Lik(1 −e−t)
et + 1

exp((y + u)t)C0(xt) =

(
∞∑
n=0

LG
(k)
n (x,y)

tn

n!

) ∞∑
j=0

uj
tj

j!




Now replacing n by n− j and comparing the coefficients of tn in both sides, we obtain (3.10).



Int. J. Anal. Appl. 16 (6) (2018) 864

Theorem 3.3. The following summation formula for Laguerre-based poly-Genocchi polynomials HG
(k)
n (z,y)

holds true:

LG
(k)
n (x + w,y + u) =

n∑
m=0


 n

m



LG

(k)
n−m(x,y)Lm(u,w). (3.11)

Proof. From (2.1) and (1.10), we have

2Lik(1 − (e)−t)
et + 1

exp((y + u)t)C0((x + w)t) =

(
∞∑
n=0

LG
(k)
n (x,y)

tn

n!

)(
∞∑
m=0

Lm(u,w)
tm

m!

)
.

Now replacing n by n−m and comparing the coefficients of tn in both sides, we get (3.11).

Theorem 3.4. The following summation formula for Laguerre-based poly-Genocchi polynomials LG
(k)
n (x,y)

holds true:

LG
(k)
n (x,y + 1) =

n∑
m=0


 n

m



LG

(k)
n−m(x,y). (3.12)

Proof. Using definition (2.1), we have

∞∑
n=0

LG
(k)
n (x,y + 1)

tn

n!
−
∞∑
n=0

LG
(k)
n (x,y)

tn

n!
=

2Lik(1 −e−t)
et + 1

exp(yt)C0(xt)(e
t − 1)

=

(
∞∑
n=0

LG
(k)
n (x,y)

tn

n!

)(
∞∑
m=0

tm

m!

)
−
∞∑
n=0

LG
(k)
n (x,y)

tn

n!

=

∞∑
n=0

n∑
m=0

LG
(k)
n−m(x,y)

tn

(n−m)!m!
−
∞∑
n=0

LG
(k)
n (x,y)

tn

n!
.

Finally, equating the coefficients of the like powers of tn, we get (3.12).

4. Identities for 2-variable Laguerre-based poly-Genocchi polynomials

In this section, we derive general symmetry identities for 2-variable Laguerre-based poly-Genocchi polyno-

mials LG
(k)
n (x,y) by applying the generating function(2.1). Such type of identities have been introduced by

several authors (see [11], [12], [13], [15]).

Theorem 4.1. Let a,b > 0 and a 6= b, x,y ∈ R, n ≥ 0, then the following identity holds true:

n∑
m=0


 n

m


an−mbmLG(k)n−m(bx,by)LG(k)m (au,aw)

=

n∑
m=0


 n

m


ambn−mLG(k)n−m(ax,ay)LG(k)m (bu,bw). (4.1)



Int. J. Anal. Appl. 16 (6) (2018) 865

Proof. Let

G(t) =

(
(2Lik(1 −e−at)(2Lik(1 −e−bt))

(eat + 1)(Abt −B−bt)

)
exp(ab(y + u)t)C0(abxt)C0(abwt). (4.2)

Since G(t) is symmetric in a and b and G(t) can written as

G(t) =

∞∑
n=0

LG
(k)
n (bx,by)

(at)n

n!

∞∑
m=0

LG
(k)
m (au,aw)

(bt)m

m!

G(t) =

∞∑
n=0


 n∑
m=0


 n

m


an−mbmLG(k)n−m(bx,by)LG(k)m (au,aw)


 tn
n!
. (4.3)

Similarly, we can show that

G(t) =

∞∑
n=0

LG
(k)
n (ax,ay)

(bt)n

n!

∞∑
m=0

LG
(k)
m (bu,bw)

(at)m

m!

G(t) =

∞∑
n=0


 n∑
m=0


 n

m


ambn−mLG(k)n−m(ax,ay)LG(k)m (bu,bw)


 tn
n!
. (4.4)

Comparing the coefficients of t
n

n!
in (4.3) and (4.4), we arrive at the desired result.

Remark 4.1. On setting b = 1 in Theorem 4.1, we get

n∑
m=0


 n

m


an−mLG(k)n−m(x,y)LG(k)m (au,aw)

=

n∑
m=0


 n

m


amLG(k)n−m(ax,ay)LG(k)m (u,w).

Theorem 4.2. Let a,b > 0 and a 6= b, x,y ∈ R and n ≥ 0, then the following identity holds true:

n∑
m=0


 n

m


a−1∑

i=0

b−1∑
j=0

LG
(k)
n−m

(
by +

b

a
i + j,bx

)
LG

(k)
m (au,aw)b

man−m

=

n∑
m=0


 n

m


 b−1∑

i=0

a−1∑
j=0

LG
(k)
n−m

(
ay +

a

b
i + j,ax

)
LG

(k)
m (bu,bw)a

mbn−m. (4.5)

Proof. Let

G(t) =

(
(2Lik(1 −e−at))(2Lik(1 −e−bt))

(eat + 1)2(ebt + 1)2

)
(eabt + 1)2 exp(ab(y + u)t)C0(abxt)C0(abwt)

G(t) =

(
2Lik(1 −e−at)

eat + 1

)
exp(abyt)C0(abxt)

(
eabt + 1

ebt + 1

)(
2Lik(1 −e−bt)

ebt + 1

)
×exp(abut)C0(abwt)

(
eabt + 1

eat + 1

)



Int. J. Anal. Appl. 16 (6) (2018) 866

=

(
2Lik(1 −e−at)

(eat + 1

)
exp(abyt)C0(abxt)

a−1∑
i=0

(−1)iebti
(

2Lik(1 −e−bt)
ebt + 1

)

×exp(abut)C0(abwt)
b−1∑
j=0

(−1)jeatj.

=

(
2Lik(1 −e−t)

eat + 1

)
C0(abxt)

a−1∑
i=0

b−1∑
j=0

(−1)i+je(by+
b
a
i+j)at

∞∑
m=0

LG
(k)
m (au,aw)

(bt)m

m!

=

∞∑
n=0

a−1∑
i=0

b−1∑
j=0

LG
(k)
n

(
by +

b

a
i + j,bx

)
(at)n

n!

∞∑
m=0

LG
(k)
m (au,aw)

(bt)m

(m)!

G(t) =

∞∑
n=0


 n∑
m=0


 n

m


a−1∑

i=0

b−1∑
j=0

(−1)i+jLG
(k)
n−m

(
by +

b

a
i + j,bx

)
LG

(k)
m (au,aw)b

man−m


 tn
n!
. (4.6)

On the other hand

G(t) =

∞∑
n=0


 n∑
m=0


 n

m


 b−1∑

i=0

a−1∑
j=0

(−1)i+jLG
(k)
n−m

(
ay +

a

b
i + j,ax

)
LG

(k)
m (bu,bw)a

mbn−m


 tn
n!
. (4.7)

On comparing the coefficients of t
n

n!
in (4.6) and (4.7), we arrive at the desired result (4.5).

Acknowledgement. All authors would like to thank Integral University, Lucknow, India, for providing the

manuscript number IU/R&D/2017-MCN000240 for the present research work.

References

[1] L. C. Andrews, Special functions for engineers and mathematicians, Macmillan Co. New York, 1985.

[2] A. Bayad and Y. Hamahata, Polylogarithms and poly-Bernoulli polynomials, Kyushu. J. Math. 65(2011), 15-24.

[3] C. H. Chang and C. W. Ha, On recurrence relation for Bernoulli and Euler numbers, Bull. Aust. Math. Soc. 64(2001),

469-474.

[4] G. S. Cheon, A note on the Bernoulli and Euler polynomials, App. Math. Lett. 16(3)(2003), 365-368.

[5] G. Dattoli, A. Torre, Operational methods and two variable Laguerre polynomials, Atti Accad. Sci. Torino Cl. Sci. Fis.

Mat. Natur. 132(1998) 3-9.

[6] G. Dattoli, A. Torre and A. M. Mancho, The generalized Laguerre polynomials, the associated Bessel functions and

applications to propagation problems, Radiat. Phys. Chem. 59(2000), 229-237.

[7] H. Jolany, M. R. Darafsheh, R. E. Alikelaye, Generalizations of Poly-Bernoulli Numbers and Polynomials, Int. J. Math.

Comb. 2(2010), 7-14.

[8] H. Jolany, R. B. Corcino, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c param-

eters, J. Class. Anal. 6(2015), 119-135.

[9] H. Jolany, M. Aliabadi, R. B. Corcino and M. R. Darafsheh, A Note on Multi Poly-Euler Numbers and Bernoulli Polyno-

mials, Gen. Math. 20(2-3)(2012), 122-134.

[10] M. Kaneko, Poly-Bernoulli numbers, J. Thor. Nombres Bordx. 9 (1997), 221-228.

[11] W. A. Khan, Some properties of the generalized Apostol type Hermite-Based polynomials, Kyungpook Math. J. 55(2015),

597-614.



Int. J. Anal. Appl. 16 (6) (2018) 867

[12] W. A. Khan, A note on Hermite-based poly-Euler and multi poly-Euler polynomials, Palestine J. Math. 6(2017), 204-214.

[13] W. A. Khan, S. Araci, M. Acikgoz, A new class of Laguerre-based Apostol type polynomials, Cogent Math. 3(2016), Art.

ID 1243839.

[14] T. Kim, Y. S. Jang and J. J. Seo, A note on poly-Genocchi numbers and polynomials, Appl. Math. Sci. 8(2014), 4475-4781.

[15] M. A. Pathan and W. A. Khan, Some implicit summation formulas and symmetric identities for the generalized Hermite-

Euler polynomials, East-West J. Math. 16(1) (2014), 92-109.

[16] H. M. Srivastava and H. L. Manocha, A treatise on generating functions Ellis Horwood Limited, New York, 1984.


	1. Introduction
	2. 2-Variable Laguerre-based poly-Genocchi polynomials
	3. Summation formulae for Laguerre-based poly-Genocchi polynomials
	4. Identities for 2-variable Laguerre-based poly-Genocchi polynomials
	References