International Journal of Analysis and Applications

Volume 17, Number 5 (2019), 803-808

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

DOI: 10.28924/2291-8639-17-2019-803

GENERALIZATION OF BATEMAN POLYNOMIALS

ASAD ALI∗, MUHAMMAD ZAFAR IQBAL, BILAL ANWER, ATHER MEHMOOD

Department of Mathematics and Statistics, University of Agriculture Faisalabad, Pakistan

∗Corresponding author:: mrasadali5100@gmail.com

Abstract. In this paper, generalize the Bateman polynomials in terms of generalized hypergeometric func-

tion of the type pFp. Establish different forms of extended polynomials such as series expansion, generating

functions and recurrence relations.

1. Introduction

Bateman polynomials are the family of Fn orthogonal polynomials. Many of researchers generalized the

classical results on the Bateman polynomials. A large dedicated literature, numbers of relevant properties,

extensions, generalizations and applications of Bateman polynomials are available in [1], [2], [4], [7], [10] and

[11]. The Bateman polynomials fn(x) generated by

∞∑
n=0

fn(x)t
n = (1 − t)−1ψ

(
−4xt

(1 − t)2

)
, (1.1)

have the following classical properties.

fn(x) =

∞∑
k=0

(−n)k(1 + n)kγkxk

( 1
2
)k(1)k

, (1.2)

fn(x) = 2F2(−n, 1 + n; 1, 1; x), (1.3)

xf
′

n(x) − nfn(x) = −nfn−1(x) − xf
′

n−1(x), n ≥ 1, (1.4)

Received 2019-04-08; accepted 2019-05-13; published 2019-09-02.

2010 Mathematics Subject Classification. 26C05, 65Q30.

Key words and phrases. Bateman polynomials; generating functions; recurrence relations.

c©2019 Authors retain the copyrights

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

803

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-803


Int. J. Anal. Appl. 17 (5) (2019) 804

xf
′

n(x) − nfn(x) = −
n−1∑
k=0

fk(x) − 2x
n−1∑
k=0

f
′

k(x), n ≥ 1, (1.5)

xf
′

n(x) − nfn(x) =
n−1∑
k=0

(−1)n−k(1 + 2k)fk(x), n ≥ 1. (1.6)

2. main results

In this section we determine generalized properties of classical Bateman polynomials, series expansion,

generating function and recurrence relations. For this let ψ(u) have a formal power- series expansion

ψ(u) =

∞∑
n=0

γnu
n, (2.1)

Define a polynomials f
(α)
n (x) by

∞∑
n=0

f(α)n (x)
n = (1 − t)−1−αψ

(
−ppxt

qq(1 − t)p

)
. (2.2)

where p ≥ 2, q = p − 1 and α is any non-negative real parameter.

Theorem 2.1. If n is non-negative integer then,

f(α)n (x) =
(1 + α)n

n!

n∑
k=0

(−n)k( 1+α+nq )k(
2+α+n

q
)k....(

q+α+n
q

)kx
kγk

( 1+α
p

)k(
2+α
p

)k....(
p+α
p

)k
. (2.3)

Proof: From (2.1) and (2.2)

∞∑
n=0

f(α)n (x)t
n =

∞∑
k=0

(−1)k(p)pkxktkγk
(q)qk(1 − t)1+α+pk

,

By using (1), pp 58 of [1]

∞∑
n=0

f(α)n (x)t
n =

∞∑
n=0

∞∑
k=0

(−1)k(p)pk(1 + α)n+pkγkxktn+k

(q)qkn!(1 + α)pk
,

By using Lemma 11, pp 57 of [1]

∞∑
n=0

f(α)n (x)t
n =

∞∑
n=0

n∑
k=0

(−1)k(p)pk(x)k(1 + α)n+qkγktn

(q)qk(n − k)!(1 + α)pk
,



Int. J. Anal. Appl. 17 (5) (2019) 805

=

∞∑
n=0

n∑
k=0

(1 + α)n
n!

(−1)kn!(p)pk(x)k(1 + α + n)qkγktn

(q)qk(n − k)!(1 + α)pk
,

equating the coefficients of tn, we obtain (2.3).

Theorem 2.2. If n ≥ 1, then

f(α)n (x) =
(1 + α)n

n!
pFp


−n, 1 + α + n

q
,

2 + α + n

q
...
q + α + n

q
; 1, 1...1︸ ︷︷ ︸
p−times

; x


 . (2.4)

Proof: If we choose

γk =
( 1+α

p
)k(

2+α
p

)k....(
p+α
p

)k

(k!)p+1
.

in (2.3) then our yield is (2.4).

Theorem 2.3. If n is non-negative integer then,

xf
′(α)
n (x) − nf

(α)
n (x) = −(α + n)f

(α)
n−1(x) − qxf

′(α)
n−1(x). (2.5)

Proof:

In order to derive (2.5), consider

F =

∞∑
n=0

f(α)n (x)t
n = (1 − t)−1−αψ(v).

where, v =
−ppxt

qq(1 − t)p
.

Differentiate with respect to x

Fx =

∞∑
n=0

f
′(α)
n (x)t

n = (1 − t)−1−αψ
′
(v)

−ppt
qq(1 − t)p

,



Int. J. Anal. Appl. 17 (5) (2019) 806

Differentiate with respect to t

Ft =

∞∑
n=0

f(α)n (x)nt
n−1 = (1 + α)(1 − t)−2−αψ(v) − (1 − t)−1−αψ

′
(v)

∂v

∂t
,

where,
∂v

∂t
=

−ppx(1 + qt)
qq(1 − t)p+1

.

Ft =

∞∑
n=0

f(α)n (x)nt
n−1 = (1 + α)(1 − t)−2−αψ(v) − x

pp(1 − t)−2−α−p(1 + qt)
qq

ψ
′
(v),

Therefore F satisfies the partial differential equation

x(1 + qt)Fx − t(1 − t)Ft + (1 + α)tF = 0.

x(1 + qt)

∞∑
n=0

f
′(α)
n (x)t

n − t(1 − t
∞∑
n=0

f(α)n (x)nt
n−1) + (1 + α)t

∞∑
n=0

f(α)n (x)t
n = 0,

∞∑
n=0

[xf
′(α)
n (x) − nf

(α)
n (x)]t

n = −
∞∑
n=0

(1 + α + n)f(α)n (x)t
n+1 − qx

∞∑
n=0

f
′(α)
n (x)t

n+1,

= −
∞∑
n=1

(α + n)f
(α)
n−1(x)t

n − qx
∞∑
n=1

f
′(α)
n−1(x)t

n,

which leads to (2.5).

Theorem 2.4. If n is non-negative integer then,

xf
′(α)
n (x) − nf

(α)
n (x) = −(1 + α)

n−1∑
k=0

f
(α)
k (x) − px

n−1∑
k=0

f
′(α)
k (x). (2.6)

Proof:



Int. J. Anal. Appl. 17 (5) (2019) 807

F also satisfies the partial differential equation

xFx − xtFx + pxtFx − tFt + t2Ft + (1 + α)tF = 0.

xFx − tFt = −
(1 + α)t

1 − t
F −

pxt

1 − t
Fx.

x

∞∑
n=0

f
′(α)
n (x)t

n − t
∞∑
n=0

f(α)n (x)nt
n−1 = −(1 + α)

∞∑
n=0

tn+1
∞∑
k=0

f
(α)
k (x)t

k − px
∞∑
n=0

tn+1
∞∑
k=0

f
′(α)
k (x)t

k,

∞∑
n=0

[xf
′(α)
n (x) − nf

(α)
n (x)]t

n = −(1 + α)
∞∑
n=0

∞∑
k=0

f
(α)
k (x)t

n+k+1 − px
∞∑
n=0

∞∑
k=0

f
′(α)
k (x)t

n+k+1

= −
∞∑
n=1

[(1 + α)

n−1∑
k=0

f
(α)
k (x) − px

n−1∑
k=0

f
′(α)
k (x)]t

n,

which leads to (2.6).

Theorem 2.5. If n is non-negative integer then,

xf
′(α)
n (x) − nf

(α)
n (x) =

n−1∑
k=0

(−q)n−k(1 + α + pk)f(α)k (x). (2.7)

Proof:

F satisfies the partial differential equation

xFx + qxtFx − tFt − qt2Ft + pt2Ft + (1 + α)tF = 0.

xFx − tFt = −
(1 + α)t

1 + qt
F −

pt2

1 + qt
Ft,

∞∑
n=0

[xf
′(α)
n (x) − nf

(α)
n (x)]t

n = −(1 + α)
∞∑
n=0

∞∑
k=0

(−q)nf(α)k (x)t
n+k+1 − p

∞∑
n=0

∞∑
k=0

(−q)nf(α)k (x)kt
n+k+1,



Int. J. Anal. Appl. 17 (5) (2019) 808

=

∞∑
n=1

n−1∑
k=0

(−q)n−k(1 + α + pk)f(α)k (x)t
n,

which gives (2.7).

For α = 0 and p = 2 the equations (2.2) to (2.7) reduces to (1.1) to (1.6).

Theorem 2.6. If n ≥ 1, then the polynomials f(α)n (x) also satisfying the following property

∞∑
n=0

f(α)n (x)t
n = (1 − t)−1−αpFp


1 + α + n

p
,

2 + α + n

p
...
q + α + n

p
; 1, 1...1︸ ︷︷ ︸
p−times

;
−ppxt
qq


 . (2.8)

Acknowledgments

The authors express their sincere gratitude to Dr. Ghulam Farid for useful discussions and invaluable advice.

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	1. Introduction
	2. main results
	References