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R E S E A R C H

Open Access

Two-parameter Srivastava polynomials and

several series identities

Cem Kaanoglu

1*

and Mehmet Ali Özarslan

2

*Correspondence:

kaanoglu@ciu.edu.tr 1Faculty of Engineering, Cyprus

International University, Mersin 10, Nicosia, Turkey

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

Abstract

In the present paper, we introduce two-parameter Srivastava polynomials in one, two and three variables by inserting new indices, where in the special cases they reduce to (among others) Laguerre, Jacobi, Bessel and Lagrange polynomials. These polynomials include the family of polynomials which were introduced and/or investigated in (Srivastava in Indian J. Math. 14:1-6, 1972; Gonzálezet al.in Math. Comput. Model. 34:133-175, 2001; Altınet al.in Integral Transforms Spec. Funct. 17(5):315-320, 2006; Srivastavaet al.in Integral Transforms Spec. Funct. 21(12):885-896, 2010; Kaanoglu and Özarslan in Math. Comput. Model. 54:625-631, 2011). We prove several two-sided linear generating relations and obtain various series identities for these polynomials. Furthermore, we exhibit some illustrative consequences of the main results for some well-known special polynomials which are contained by the two-parameter Srivastava polynomials.

MSC: 33C45

1 Introduction

Let{An,k}∞n,k=be a bounded double sequence of real or complex numbers, let [a] denote

the greatest integer ina∈R, and let (λ)ν, (λ)≡, denote the Pochhammer symbol

de-fined by

(λ)ν:=(λ+ν)

(λ)

by means of familiar gamma functions. In , Srivastava [] introduced the following family of polynomials:

SNn(z) :=

[Nn]

k=

(–n)Nk

k! An,kz

k nN

=N∪ {};N∈N

, ()

whereNis the set of positive integers.

Afterward, Gonzálezet al.[] extended the Srivastava polynomialsSN

n(z) as follows:

SNn,m(z) :=

[Nn]

k=

(–n)Nk

k! An+m,kz

k (m,nN

;N∈N) ()

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and investigated their properties extensively. Motivated essentially by the definitions () and (), scientists investigated and studied various classes of Srivastava polynomials in one and more variables.

In [], the following family of bivariate polynomials was introduced:

Smn,N(x,y) :=

[Nn]

k=

Am+n,k

xnNk (n–Nk)!

yk

k! (n,m∈N,N∈N),

and it was shown that the polynomialsSm,N

n (x,y) include many well-known polynomials

such as Lagrange-Hermite polynomials, Lagrange polynomials and Hermite-Kampé de Feriét polynomials.

In [], Srivastavaet al.introduced the three-variable polynomials

Smn,M,N(x,y,z)

:=

[Nn]

k= [Mk]

l=

Am+n,k,l

xl

l! ykMl

(k–Ml)! znNk

(n–Nk)! (m,n∈N;M,N∈N), ()

where {Am,n,k}is a triple sequence of complex numbers. Suitable choices of{Am,n,k} in

equation () give a three-variable version of well-known polynomials (see also []). Re-cently, in [], the multivariable extension of the Srivastava polynomials inr-variable was introduced

Smn,N,N,...,Nr–(x,x, . . . ,xr)

:=

[Nr–n ]

kr–= [Nr–kr–]

kr–=

· · ·

[kN]

k= [k

N]

k=

Am+n,kr–,k,k,...,kr–

xk k!

xkNk (k–Nk)!· · ·

xnrNr–kr–

(n–Nr–kr–)!

(m,n∈N;N,N, . . . ,Nr–∈N), ()

where{Am,kr–,k,k,...,kr–}is a sequence of complex numbers.

In this paper we introduce the two-parameter Srivastava polynomials in one and more variables by inserting new indices. These polynomials include the family of polynomials which were introduced and/or investigated in [–, , ] and []. We prove several two-sided linear generating relations and obtain various series identities for these polynomials. Furthermore, we exhibit some illustrative consequences of the main results for some well-known special polynomials which are contained by the two-parameter Srivastava polyno-mials.

2 Two-parameter one-variable Srivastava polynomials

In this section we introduce the following family of two-parameter one-variable polyno-mials:

Smn ,m(x) :=

n

k=

(–n)k

k! Am+m+n,m+kx

k (m

(3)

where{An,k}is a bounded double sequence of real or complex numbers. Note that

appro-priate choices of the sequenceAn,kgive one-variable versions of the well-known

polyno-mials.

Remark . ChoosingAm,n= (–αm)n(m,n∈N) in (), we get

Smn ,m

– x

= (–)m(α+m+n+ )m

n! (–x)nL

(α+m)

n (x),

whereL()(x) are the classical Laguerre polynomials given by

L(α)

n (x) =

(–x)n

n!F

–n, –αn; –;– x

.

Remark . Setting

Am,n=

(α+β+ )m(–βm)n

(α+β+ )m(–αβ– m)n

(m,n∈N)

in (), we obtain

Smn ,m

  +x

= (α+β+ )m+m+n(–βm–m–n)m( +α+β+ m+m)n (α+β+ )m+m+n(–αβ– m– m– n)m( +α+β+ m+m)n

×n!

  +x

n

P(nα+m+m,β+m)(x),

wherePn(α,β)(x) are the classical Jacobi polynomials.

Remark . If we setAm,n= (α+m– )n(m,n∈N) in (), then we get

Sm,m n

–x

β

= (α+m+m+n– )myn(x,α+m+ m,β) (β= ),

whereyn(x,α,β) are the classical Bessel polynomials given by

yn(x,α,β) =F

–n,α+n– ; –;–x

β

.

Theorem . Let{f(n)}∞n=be a bounded sequence of complex numbers.Then

m,m,n=

f(n+m+m)Smn ,m(x)

wm m!

wm m!

tn

n!

= ∞

m,m=

f(m+m)Am+m,m

(w+t)m

m!

(w+ (–xt))m

m!

, ()

(4)
(5)

Remark . Setting

Am,n=

(α+β+ )m(–βm)n

(α+β+ )m(–αβ– m)n

(m,n∈N)

andx+xin equation (), we have

m,m,n=

f(n+m+m)

× (α+β+ )m+m+n(–βm–m–n)m( +α+β+ m+m)n

(α+β+ )m+m+n(–αβ– m– m– n)m( +α+β+ m+m)n

×n!

  +x

n

P(nα+m+m,β+m)(x)w

m

m!

wm m!

tn

= ∞

m,m=

f(m+m)

(α+β+ )m+m(–βm–m)m

(α+β+ )m+m(–αβ– m– m)m

×(w+t)m

m!

(w+ (–+xt))m

m!

.

Remark . If we setAm,n= (α+m– )n(m,n∈N) andx→–βx in (), then we can write

m,m,n=

f(n+m+m)(α+m+m+n– )myn(x,α+m+ m,β)

wm m!

wm m!

tn

n!

= ∞

m,m=

f(m+m)(α+m+m– )m

(w+t)m

m!

(w+xβt)m

m!

.

If we setf=  andw= –βxt, then

m,m,n=

(α+m+m+n– )myn(x,α+m+ m,β)

wm m!

(–βxt)m

m!

tn

n!

= ∞

m=

(w+t)m

m!

=ew+t.

Remark . If we consider Remarks . and ., we get the following relation between

Laguerre polynomialsL()(x) and Bessel polynomialsyn(x,α,β):

m,m,n=

(–)m(α+m+n+ )mL(+m)(x)

wm m!

(–tx)n+m m!

= ∞

m,m,n=

(α+m+m+n– )myn(x,α+m+ m,β)

wm m!

(–βxt)m

m!

tn

(6)

3 Two-parameter two-variable Srivastava polynomials

In this section we introduce the following two-parameter family of bivariate polynomials:

Smn ,m(x,y) :=

n

k=

Am+m+n,m+k

xk

k! ynk

(n–k)! (m,m,n,k∈N), ()

where{An,k}is a bounded double sequence of real or complex numbers. Note that in the

particular case these polynomials include the Lagrange polynomials.

Remark . ChoosingAm,n= (α)mn(β)n(m,n∈N) in (), we have

Smn ,m(x,y) = (α)m(β)mgn(α+m,β+m)(x,y),

wheregn(α,β)(x,y) are the Lagrange polynomials given by

gn(α,β)(x,y) =

n

k=

(α)nk(β)k

(n–k)!k! x

kynk.

Using similar techniques as in the proof of Theorem ., we get the following theorem.

Theorem . Let{f(n)}∞n=be a bounded sequence of complex numbers.Then

m,m,n=

f(n+m+m)Smn ,m(x,y)

wm m!

wm m!

tn

= ∞

m,m=

f(m+m)Am+m,m

(w+yt)m

m!

(w+xt)m

m!

, ()

provided each member of the series identity()exists.

Remark . If we setAm,n= (α)mn(β)n(m,n∈N) in (), we have

m,m,n=

f(n+m+m)(α)m(β)mgn(α+m,β+m)(x,y)

wm m!

wm m!

tn

= ∞

m,m=

f(m+m)(α)m(β)m

(w+yt)m

m!

(w+xt)m

m!

.

Choosingf=  gives

m,m,n=

(α)m(β)mgn(α+m,β+m)(x,y)

wm m!

wm m!

tn

= ( –w–yt)α( –w–xt)β.

Furthermore, since we have the relation between P(,β)(x,y) Jacobi polynomials and

Lagrange polynomials [] as

g(α,β)

n (x,y) = (yx)nPn(–αn,–βn)

x+y xy

(7)

we get the following generating relation for Jacobi polynomialsP(,β)(x,y):

m,m,n=

(α)m(β)m(y–x)nP(–mn,–βmn)

x+y xy

wm m!

wm m!

tn

= ( –w–yt)α( –w–xt)β.

4 Two-parameter three-variable Srivastava polynomials

In this section we define two-parameter three-variable Srivastava polynomials as follows:

Smn ,m,M(x,y,z) :=

n

k= [Mk]

l=

Am+m+n,m+k,l

xl

l! ykMl

(k–Ml)! znk

(n–k)!

(m,m,n,k,l∈N,M∈N), ()

where{An,k,l}∞n,k=is a bounded triple sequence of real or complex numbers.

Using similar techniques as in the proof of Theorem ., we get the following theorem.

Theorem . Let{f(n)}∞n=be a bounded sequence of complex numbers.Then

m,m,n=

f(n+m+m)Snm,m,M(x,y,z)

wm m!

wm m!

tn

= ∞

m,m,l=

f(m+m+Ml)Am+m+Ml,m+Ml,l

(xtM)l l!

(w+zt)m

m!

(w+yt)m

m!

, ()

provided each member of the series identity()exists.

Theorem . Let{f(n)}∞n=be a bounded sequence of complex numbers,and let Smn ,m,M(x,

y,z)be defined by().Suppose also that two-parameter two-variable polynomials PM m,m(x,

y)are defined by

PMm,m(x,y) =

[mM]

l=

Am+m,m,l

xmMl

(m–Ml)!

yl

l!. ()

Then the family of sided linear generating relations holds true between the two-parameter three-variable Srivastava polynomials Smn ,m,M(x,y,z)and PmM,m(x,y):

m,m,n=

f(n+m+m)Snm,m,M(x,y,z)

wm m!

wm m!

tn

= ∞

m,m=

f(m+m)

(w+zt)m

m!

PMm,mw+yt,xtM

. ()

Suitable choices ofAn,k,lin equations () and () give some known polynomials. Remark . ChoosingM=  andAm,n,k= (α)mn(γ)n–k(β)k(m,n∈N) in (), we get

(8)

whereu(,β,γ)(x,y,z) is the polynomial given by

u(nα,β,γ)(x,y,z) =

n

k= [k]

l=

(β)l(γ)k–l(α)nk

yl

l! xnk

(n–k)! zk–l

(k– l)!.

Now, by settingM=  andAm,n,k= (α)k(β)nk(γ)mn(m,n∈N) in the definition (), we

obtain

Smn ,m,(x,y,z) = (γ)m(β)mgn(α,β+m,γ+m)(x,y,z),

wheregn(α,β,γ)(x,y,z) are the Lagrange polynomials given by

gn(α,β,γ)(x,y,z) =

n

k=

k

l=

(α)l(β)kl(γ)nk

xl l!

ykl (k–l)!

znk (n–k)!.

Remark . If we setM=  andAm,n,k= (α)mn(γ)n–k(β)k(m,n∈N) in (), then

Pm,m(x,y) = (α)mh(mγ,β)(x,y),

whereh(mγ,β)(x,y) denotes the Lagrange-Hermite polynomials given explicitly by

h(mγ,β)(x,y) =

[m  ]

l=

(γ)m–l(β)l

xm–l

(m– l)!

yl

l!.

Furthermore, choosingM=  andAm,n,k= (α)k(β)nk(γ)mn(m,n∈N) in the definition

(), we have

Pm,m(x,y) = (γ)mgm(β,α)(x,y),

wheregm(α,β)(x,y) are the Lagrange polynomials given by

g(α,β)

m (x,y) = m

l=

(α)ml(β)l

xml

(m–l)!

yl

l!.

Remark . If we setw→–ztandw→–ytin Theorem ., then we get

m,m,n=

f(n+m+m)Snm,m,M(x,y,z)

(–zt)m

m!

(–yt)m

m!

tn

= ∞

l=

f(Ml)AMl,Ml,l

(xtM)l

l! . ()

Now, if we setf = ,M=  andAm,n,k= (α)mn(γ)n–k(β)k(m,n∈N) in (), then

m,m,n=

(α)m(γ)mun(α+m,β,γ+m)(z,x,y)

(–zt)m

m!

(–yt)m

m!

(9)

Furthermore, if we setM=  andAm,n,k= (α)mn(γ)n–k(β)k(m,n∈N) in (), then

m,m,n=

f(n+m+m)(α)m(γ)mu(+m,β,γ+m)(z,x,y)

wm

m!

wm

m!

tn

= ∞

m,m=

f(m+m)

(w+zt)m

m!

(α)mh(mγ,β)

w+yt,xt

.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors completed the paper together. Both authors read and approved the final manuscript.

Author details

1Faculty of Engineering, Cyprus International University, Mersin 10, Nicosia, Turkey.2Department of Mathematics, Faculty

of Arts and Sciences, Eastern Mediterranean University, Mersin 10, Gazimagusa, Turkey.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

We would like to thank the referees for their valuable comments.

Received: 12 December 2012 Accepted: 6 March 2013 Published: 29 March 2013 References

1. Srivastava, HM: A contour integral involving Fox’sH-function. Indian J. Math.14, 1-6 (1972)

2. González, B, Matera, J, Srivastava, HM: Someq-generating functions and associated generalized hypergeometric polynomials. Math. Comput. Model.34, 133-175 (2001)

3. Altın, A, Erku¸s, E, Özarslan, MA: Families of linear generating functions for polynomials in two variables. Integral Transforms Spec. Funct.17(5), 315-320 (2006)

4. Srivastava, HM, Özarslan, MA, Kaanoglu, C: Some families of generating functions for a certain class of three-variable polynomials. Integral Transforms Spec. Funct.21(12), 885-896 (2010)

5. Kaanoglu, C, Özarslan, MA: New families of generating functions for certain class of three-variable polynomials. Appl. Math. Comput.218, 836-842 (2011)

6. Kaanoglu, C, Özarslan, MA: Two-sided generating functions for certain class ofr-variable polynomials. Math. Comput. Model.54, 625-631 (2011)

7. Özarslan, MA: Some families of generating functions for the extended Srivastava polynomials. Appl. Math. Comput.

218, 959-964 (2011)

8. Liu, S-J, Chyan, C-J, Lu, H-C, Srivastava, HM: Multiple integral representations for some families of hypergeometric and other polynomials. Math. Comput. Model.54, 1420-1427 (2011)

9. Srivastava, HM, Manocha, HL: A Treatise on Generating Functions. Halsted, New York (1984)

doi:10.1186/1687-1847-2013-81

References

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