On σ type zero of Sheffer polynomials

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R E S E A R C H

Open Access

On

σ

-type zero of Sheffer polynomials

Ajay K Shukla

*

, Shrinivas J Rapeli and Pratik V Shah

*Correspondence: ajayshukla2@rediffmail.com Department of Mathematics, S. V. National Institute of Technology, Surat, 395 007, India

Abstract

The main object of this paper is to investigate some properties of

σ

-type polynomials in one and two variables.

MSC: 33C65; 33E99; 44A45; 46G25

Keywords: Appell sets; differential operator; Sheffer polynomials; generalized Sheffer polynomials

1 Introduction

In , Thorne [] obtained an interesting characterization of Appell polynomials by means of the Stieltjes integral. Srivastava and Manocha [] discussed the Appell sets and polynomials. Dattoliet al.[] studied the properties of the Sheffer polynomials. Recently Pintér and Srivastava [] gave addition theorems for the Appell polynomials and the as-sociated classes of polynomial expansions and some cases have also been discussed by Srivastava and Choi [] in their book.

Appell sets may be defined by the following equivalent condition:{Pn(x)},n= , , , . . . is an Appell set [–] (Pnbeing of degree exactlyn) if either

(i) Pn(x) =Pn–(x),n= , , , . . ., or

(ii) there exists a formal power seriesA(t) =∞n=antn(a= ) such that

A(t)exp(xt) =

n=

Pn(x)tn.

Sheffer’s A-type classification

Letφn(x) be a simple set of polynomials and letφn(x) belong to the operator

J(x,D) =

k=

Tk(x)Dk+,

withTk(x) of degree≤k. If the maximum degree of the coefficientsTk(x) ism, then the setφn(x) is of Sheffer A-typem. If the degree ofTk(x) is unbounded ask→ ∞, we say that φn(x) is of Sheffer A-type∞.

Polynomials of Sheffer A-type zero

Letφn(x) be of Sheffer A-type zero. Thenφn(x) belong to the operator

J(D) =

k=

ckDk+,

(2)

in whichckare constants. Herec=  andJφn=φn–. Furthermore, sinceckare indepen-dent ofxfor everyk, a functionJ(t) exists with the formal power series expansion

J(t) =

k=

cktk+, c= .

LetH(t) be the formal inverse ofJ(t); that is,

HJ(t)=JH(t)=t.

Theorem(Rainville []) A necessary and sufficient condition thatφn(x)be of Sheffer A-type zero is thatφn(x)possess the generating function indicated in

A(t)expxH(t)=

n=

φn(x)tn,

in which H(t)and A(t)have(formal)expansions

H(t) =

n=

hntn+, h= , A(t) =

n=

antn, a= .

Theorem(Al-Salam and Verma []) Let{Pn(x)}be a polynomial set.In order for{Pn(x)}

to be a Sheffer A-type zero,it is necessary and sufficient that there exist(formal)power series

H(t) =

j=

hjtj, h= , As(t) =

j=

a(js)tj not all a(s)are zero

and

r

j=

Aj(t)expxHjt)

=

n=

Pn(x)tn,

where

J(D)Pn(x) =Pnr(x) (n=r,r+ , . . .)where J(D) =

k=

akDk+r,a= 

and r is a fixed positive integer.The function A(t)may be called the determining function for the set{Pn(x)}.

Polynomial of

σ

-type zero [9, 11]

Let{pn(x)}be a simple set of polynomials that belongs to the operator

J(x,σ) =

k=

Tk(xk+,

σ=D

q

i=

(xD+bi– ), D=

d dx,

J(x,σ)pn(x) =pn–(x)

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wherebiare constants, not equal to zero or a negative integer, andTk(x) are polynomials of degree≤k. We can say that this set is ofσ-typemif the maximum degree ofTk(x) is

m,m= , , , . . . .

A necessary and sufficient condition thatφn(x) be ofσ-type zero, with

σ=D

q

i=

(xD+bi– ),

is thatφn(x) possess the generating function

A(t)Fq

–;b,b, . . . ,bq;xH(t)

=

n=

φn(x)tn,

in whichH(t) andA(t) have (formal) expansions

H(t) =

n=

hntn+, h= ,

and

A(t) =

n=

antn, a= .

Sinceφn(x) belongs to the operatorJ(σ) = ∞

k=ckσk+, whereckare constant andc= .

2 Main results

Theorem  If pn(x)is a polynomial set,then pn(x)is ofσ-type zero withσ=Dqm=(xD+

bm– ).It is necessary and sufficient condition that there exist formal power series

H(t) =

n=

hntn+, h= ,

and

Ai(t) =

n=

a(ni)tn not all a(i)are zero

such that

r

i=

Ai(t)Fq

–;b,b, . . . ,bq;xHit)

=

n=

pn(x)tn, ()

whereθ=xD.

Proof Letyi=Fq(–;b,b, . . . ,bq;zi), where i= , , . . . ,r, be a solution of the following

differential equation:

θ q

m=

(xD+bm– ) –zi yi= , θ=xD, D=

(4)

On substitutingzi=xHit) and keepingtas a constant, where

σ=D

q

m=

(xD+bm– ), θ=xD,

we get

xD

q

m=

(θ+bm– ) –xH(zi) yi= .

This can also be written as

σyi=Hit)yi

or

σFq

–;b,b, . . . ,bq;xHit)

=Hit)Fq

–;b,b, . . . ,bq;xHit)

.

OperatingJ(σ) on both sides of Equation () yields

J(σ)

n=

pn(x)tn=J(σ) r

i=

Ai(t)Fq

–;b,b, . . . ,bq;xHit)

= r

i=

Ai(t)JHit)

Fq

–;b,b, . . . ,bq;xHit)

=t

n=

pn(x)tn

=

n=

pn–(x)tn.

Therefore,J(σ)p(x) =  andJ(σ)pn(x) =pn–(x),n≥.

SinceJ(σ) is independent ofx, using the definition ofσ-type [, ], we arrive at the conclusion thatpn(x) isσ-type zero.

Conversely, supposepn(x) is ofσ-type zero and belongs to the operatorJ(σ). Nowqn(x) is a simple set of polynomials, we can write

r

i= Fq

–;b,b, . . . ,bq;xHit)

=

n=

pn(x)tn, ()

whereε,ε, . . . ,εrare the roots of unity.

Sinceqn(x) is a simple set, there exists a sequenceck[], independent ofn, such that

pn(x) = n

k=

(5)

and

n=

pn(x)tn=

n=

n

k=

cnkqk(x)tn.

On replacingnbyn+k, this becomes

n=

pn(x)tn=

n=

k=

cnqk(x)tn+k

=

k=

qk(x)tk

n=

cntn.

Settingcn=a(ni)(iis independent ofn, wherei= , , . . . ,r), this becomes

n=

pn(x)tn=

k=

qk(x)tk

n=

a(ni)tn, by using Equation (), we get

= r

i=

Ai(t)Fq

–;b,b, . . . ,bq;xHit)

.

This completes the proof.

Theorem  A necessary and sufficient condition that pn(x)be ofσ-type zero and there

exist a sequence hk,independent of x and n,such that

r

i=

εnihn–ψ(εit) =σpn(x), ()

whereψit) =Ai(t)Fq(–;b,b, . . . ,bq;xHit)).

Proof Ifpn(x) is ofσ-type zero, then it follows from Theorem  that

n=

pn(x)tn= r

i=

Ai(t)Fq

–;b,b, . . . ,bq;xHit)

.

This can be written as

n=

σpn(x)tn= r

i=

Ai(t)σFq

–;b,b, . . . ,bq;xH(εit)

= r

i=

Hit)Ai(t)Fq

–;b,b, . . . ,bq;xHit)

= r

i=

n=

hnεin+tn+

Ai(t)Fq

–;b,b, . . . ,bq;xHit)

=

n=

r

i=

εnihn–

(6)

Thus

σpn(x) = r

i=

εnihn–ψit).

This completes the proof.

3 Sheffer polynomials in two variables [12]

Letpn(x,y) be ofσ-type zero. Thenpn(x,y) belongs to an operatorJ(σ) = ∞

k=ckσk+, in whichckare constants andc= .

Since

J(σ)pn(x,y) =pn–(x,y), n≥,

where

Dx=

∂x, Dy=

∂y, θ=x

∂x, φ=y

∂y,

σx=Dx p

m=

(θ+bm– ), σy=Dy q

s=

(θ+bs– ),

and

J(G+H)(t)=(G+H)J(t)=t, σ=σx+σy.

Theorem  A necessary and sufficient condition that pn(x,y)be ofσ-type zero,with

σx=Dx p

m=

(θ+bm– ), σy=Dy q

s=

(θ+bs– ), σ=σx+σy,

is that pn(x,y)possess a generating function in

r

i=

Ai(t)Fp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

=

n=

pn(x,y)tn, ()

in which

G(t) =

n=

gntn+, g= ,

H(t) =

n=

hntn+, h= ,

Ai(t) =

n=

a(ni)tn not all a(i)are zero

(7)

Proof Letui=Fp(–;b,b, . . . ,bp;zi) andvi=Fq(–;c,c, . . . ,cq;wi) be the solutions of the

following differential equations:

θz

p

m=

z+bm– ) –zi ui= , θz=z ∂z,

and

φw q

s=

(φw+cs– ) –zi wi= , φw=w ∂w.

On substitutingzi=xGit),wi=yHit) and keepingtas a constant, whereθ=xx=θz, φ=y∂

∂y=φw, we get

θ p

m=

(θ+bm– )ui=xGit)ui

and

φ q

s=

(φ+cs– )wi=yHit)wi.

This can also be written as

σFp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

=G(εit) +H(εit)Fp

–;b,b, . . . ,bp;xG(εit)

Fq

–;c,c, . . . ,cq;yH(εit)

.

OperatingJ(σ) on both sides of Equation () yields

J(σ)

n=

pn(x,y)tn

=J(σ) r

i=

Ai(t)Fp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

= r

i=

Ai(t)J(G+H)(εit)

Fp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

=t

n=

pn(x,y)tn

=

n=

pn–(x,y)tn.

Therefore,J(σ)p(x,y) =  andJ(σ)pn(x,y) =pn–(x,y),n≥.

(8)

Conversely, supposepn(x,y) is ofσ-type zero and belongs to the operatorJ(σ). Now

qn(x,y) is a simple set of polynomials. We can write

r

i= Fp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

=

n=

pn(x,y)tn. ()

Sinceqn(x,y) is a simple set, there exists a sequenceck, independent ofn, such that

pn(x,y) = n

k=

cnkqk(x,y)

and

n=

pn(x,y)tn=

n=

n

k=

cnkqk(x,y)tn.

On replacingnbyn+k, this becomes

=

n=

k=

cnqk(x,y)tn+k

=

k=

qk(x,y)tk

n=

cntn.

Settingcn=a(ni)(iis independent ofn, wherei= , , . . . ,r), this becomes

=

k=

qk(x,y)tk

n=

a(ni)tn

= r

i=

Ai(t)Fp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

.

This completes the proof.

Theorem  A necessary and sufficient condition that pn(x,y)be ofσ-type zero and there exist sequences gkand hk,independent of x,y and n,such that

r

i=

εni(gn–+hn–)υ(εit) =σpn(x,y), ()

whereυit) =Ai(t)Fp(–;b,b, . . . ,bp;xGit))Fq(–;c,c, . . . ,cq;yHit)).

Proof Ifpn(x,y) is ofσ-type zero, then it follows from Theorem  that

n=

pn(x,y)tn= r

i=

Ai(t)Fp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

(9)

This can be written as

n=

σpn(x,y)tn= r

i=

Ai(t)σFp

–;b,b, . . . ,bp;xGit)

Fq

–;c,c, . . . ,cq;yHit)

= r

i=

(G+H)(εit)Ai(t)Fp

–;b,b, . . . ,bp;xGit)

×Fq

–;c,c, . . . ,cq;yHit)

= r

i=

n=

(gn+hnin+tn

+

Ai(t)Fp

–;b,b, . . . ,bp;xGit)

×Fq

–;c,c, . . . ,cq;yHit)

=

n=

r

i=

εni(gn–+hn–)

υ(εit)tn.

Thus

σpn(x,y) = r

i=

εni(gn–+hn–)υ(εit),

whereυ(εit) =Ai(t)Fp(–;b,b, . . . ,bp;xGit))Fq(–;c,c, . . . ,cq;yHit)). This completes

the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this article. The authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors are indebted to the referees for their valuable suggestions which led to a better presentation of paper.

Received: 29 January 2013 Accepted: 19 April 2013 Published: 14 May 2013

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(2009)

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doi:10.1186/1029-242X-2013-241

Cite this article as:Shukla et al.:Onσ-type zero of Sheffer polynomials.Journal of Inequalities and Applications2013

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