Ghazala Yasmin

Department of Applied Mathematics,Faculty of Engineering,Aligarh Muslim University,Aligarh,India

Abstract.In this article,the 2-variable general polynomials are taken as base with Peters polynomials to introduce a family of 2-variable Peters mixed type polynomials.These polynomials are framed within the context of monomiality principle and their properties are established.Certain summation formulae for these polynomials are also derived.Examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored.

KeyWords:2-Variablegeneralpolynomials,Peterspolynomials,2-variabletruncatedexponential polynomials,Sheffer sequences,monomiality principle.

1 Introduction and preliminaries

Special functions(or special polynomials)because of their remarkable properties known as ”useful functions” and have been used for centuries.In the past years,the development of new special functions and of applications of special functions to new areas of mathematics have initiated a revival of interest in the p-adic analysis,q-analysis,analytic number theory,combinatorics and so on.Moreover,in recent years,the generalized and multi-variable forms of the special functions of mathematical physics have witnessed a significant evolution.In particular,the special polynomials of two variables provided new means of analysis for the solution of large classes of partial differential equations often encountered in physical problems see for example[3,4,6,7,11].Most of the special functions of mathematical physics and their generalizations have been suggested by physical problems.Also these polynomials are helpful in introducing new families of special polynomials.

Motivated by the importance of the special functions of two variables in applications,a general class of the 2-variable polynomials,namely the 2-variable general polynomials(2VGP)pn(x,y)is considered in[11].These polynomials are defined by the generating function[11,p.4(14)]

Table 1:List of special cases of 2VGP pn(x,y).

where φ(y,t)has(at least the formal)series expansion

The 2VGP family contains very important polynomials such as the Gould-Hopper polynomials(GHP)(x,y),2-variable Hermite Kampé de Feriet polynomials(2VHKdFP)Hn(x,y),2-variable generalized Laguerre polynomials(2VGLP)mLn(y,x),2-variable Laguerre polynomials(2VLP)Ln(y,x),2-variable truncated exponential polynomials(of order r)(2VTEP)(x,y)and 2-variable truncated exponential polynomials(2VTEP)[2]en(x,y).We present the list of some known 2VGP family in Table 1.

It is shown in[11],that the polynomials pn(x,y)are quasi-monomial[4,19]with respect to the following multiplicative and derivative operators:

and

respectively.

According to the monomiality principle and in view of Eqs.(1.3)and(1.4),we have

and

respectively.

Now,since the 2VGP pn(x,y)are quasi-monomial,the properties of these polynomials can be derived from those of the multiplicative and derivative operatorsandrespectively.In fact,we have

which yields the following differential equation satisfied by pn(x,y):

Again,since p0(x,y)=1,the 2VGP pn(x,y)can be explicitly constructed as:

Identity(1.9)implies that the exponential generating function of the 2VGP pn(x,y)can be cast in the form

which yields generating function(1.1).Again,expanding the exponential function extand using series expansion(1.2)in the l.h.s.of generating function(1.1),we get the following series definition of the 2VGP pn(x,y):

It can easily be verified that theandsatisfy the following commutation relation:

Sequences of polynomials play an important role in various branches of sciences.One of the important classes of polynomial sequences is the class of Sheffer sequences.It is a polynomial sequence in which the index of each polynomial equal its degree,satisfying some conditions related to the umbral calculus in combinatorics.There are several ways to define the Sheffer sequences[18],among which by a generating function and by a differential recurrence relation are most common.Sheffer polynomials are classified and many important properties are derived by Rainville[16].Roman[17]studied these Sheffer polynomials and handled its properties naturally within the framework of modern and classical umbral calculus.In view of this approach many wonderful results are derived for combinatorics theory,see for example[9,15].

The Peters polynomials(PP)Sn(x)and their related polynomials Boole polynomials(BlP)Bln(x)and the Changhee polynomials(ChP)Chn(x),belong to the class of Sheffer sequences.The Peters polynomials Sn(x)plays an important role in the area of number theory,algebra and umbral calculus and are defined by means of the generating function[12]

The first few Peters polynomials are given by[12]

In particular,we note that

where Bln(x;λ)denotes the Boole polynomials(BlP)[13]and Chn(x)denotes the Changhee polynomials(ChP)[14]defined by

and

Taking x=0 in generating functions(1.13),(1.16)and(1.17),we find

and

where

are the corresponding numbers.

The Stirling number of the first kind is given by

Thus,by(1.21),we get

It is easy to show that

Motivated by the work done on mixed type polynomials and due to the importance of the 2-variable forms of the special polynomials in this paper,a family of the 2-variable Peters mixed type polynomials is introduced by means of generating function and series definition.Certain properties and summation formulae for these polynomials are derived.Examples of some members belonging to this family are considered.The numbers corresponding to certain mixed special polynomials are also explored.

2 2-variable Peters mixed type polynomials

In this section,we introduce the 2-variable Peters mixed type polynomials(2VPmTP)by means of generating function and series definition.Further,we also derive certain properties and summation formulae for these polynomials.In order to derive the generating function for the 2VPmTP,we prove the following result:

Theorem 2.1.The generating function for the 2-variable Peters mixed type polynomials 2VPmTPpSn(x,y;λ;μ)is given as:

Proof.Replacing x in the l.h.s.and r.h.s.of generating function(1.13)by the multiplicative operatorof the 2VGP pn(x,y),we have

Using Eq.(1.10)in the l.h.s.and Eq.(1.3)in the r.h.s.of Eq.(2.2),we find

Now,using Eq.(1.1)in the l.h.s.and denoting the resultant 2-variable Peters mixed type polynomials(2VPmTP)in the r.h.s.bypSn(x,y;λ;μ),that is

we get assertion(2.1).

The 2-variable Peters mixed type polynomials(2VPmTP),denoted bypSn(x,y;λ;μ)will be defined as the discrete Peters convolution of the 2-variable general polynomials pn(x,y).

Remark 2.1.We remark that Eq.(2.4)gives the operational representation between the Peters polynomials Sn(x;λ;μ)and 2VPmTPpSn(x,y;λ;μ).

Next,we obtain the series definition of the 2VPmTPpSn(x,y;λ;μ)by proving the following result:

Theorem 2.2.The 2-variable Peters mixed type polynomials 2VPmTPpSn(x,y;λ;μ)are defined by the series:

Proof.Using expansion(1.18a)and Eq.(1.22)in the l.h.s.of Eq.(2.3)and Eq.(2.4)in the r.h.s.of Eq.(2.3),we find

Replacing n by n?q in the l.h.s.of Eq.(2.6)and then equating the coefficients of like powers of t in both sides of the resultant equation,we get assertion(2.5).

Note that from definition(2.5),we conclude that the 2VPmTPpSn(x,y;λ;μ)are defined as the discrete Peters convolution of the 2-variable general polynomials pn(x,y).

Further,to frame the 2VPmTPpSn(x,y;λ;μ)within the context of monomiality principle,we prove the following result:

Theorem 2.3.The 2VPmTPpSn(x,y;λ;μ)are quasi-monomial with respect to the following multiplicative and derivative operators:

and

respectively.

Proof.Consider the identity

Differentiating Eq.(2.1)partially with respect to t,we find

Since φ(y,t)is an invertible series of t,thereforep ossess power series expansion of t.Thus,in view of the identity(2.9),the above equation becomes

Now,using generating relation(2.1)in the l.h.s.of Eq.(2.11),rearranging the summation and equating the coefficients of the same powers of t in both sides of the resultant equation,we find

which in view of monomiality principle equation(1.5)(forpSn(x,y;λ;μ))yields assertion(2.7).

Now,to prove assertion(2.8),we note that in view of generating function(2.1)and identity(2.9),we have

Table 2:Special cases of the 2VPmTPpSn(x,y;λ;μ).

Rearrang ing the summa tioninthel.h.s.ofEq.(2.13)and then equatingthecoe fficients of the same powers of t in both sides of the resultant equation,we find

which in view of monomiality principle equation(1.6)(forpSn(x,y;λ;μ))yields assertion(2.8).

We know that the properties of quasi-monomial can be derived by using the expressions of the multiplicative and derivative operators.To derive the differential equation for the 2VPmTPpSn(x,y;λ;μ)we prove the following result:

Theorem 2.4.The 2VPmTPpSn(x,y;λ,μ)satisfy the following differential equation:

Proof.Using expressions(2.7)and(2.8)and in view of monomiality principle equation(1.7),we get assertion(2.15).

Remark 2.2.We remark that Eqs.(2.12)and(2.14)are the differential recurrence relations satisfied by the 2VPmTPpSn(x,y;λ;μ).

By taking suitable values of the parameters in Eqs.(2.1),(2.4),(2.5),(2.7),(2.8),(2.15)and in view of relations(1.15a)and(1.15b),we can find the generating functions and other results for the mixed special polynomials related topSn(x,y;λ;μ).We present the generating functions and series definitions for these polynomials in Table 2.

It happens very often that the solution of a given problem in physics or applied mathematics requires the evaluation of in finite sums involving special functions.The summation formulae of special functions of more than one variable often appear in applications ranging from electromagnetic processes to combinatorics,see for example[5].The importance of summation formulae of special functions provides motivation to find the summation formulae for the 2VPmTPpSn(x,y;λ;μ).

We derive the summation formulae for the 2VPmTPpS(x,y;λ;μ)in the form of following theorems:

Theorem 2.5.The following implicit summation formula for the 2VPmTPpSn(x,y;λ;μ)holds true:

Proof.Replacing x→x+w in generating function(2.1),we have

Again,using Eq.(2.1)and series expansion of ewtin the l.h.s.of Eq.(2.17),we find

which on replacing n by n?k in the l.h.s.and then equating the coefficients of the same powers of t in both sides of the resultant equation yields assertion(2.16).

Remark2.3.Weremarkthat,usingEqs.(1.1),(1.13)and(1.22)inthel.h.s.ofEq.(2.17)and then applying the Cauchy-product rule in the resultant equation,we obtain the following explicit summation formula for the Peters polynomials Sn(x;λ;μ)in terms of the 2VPmTPpSn(x,y;λ;μ):

Theorem 2.6.The following implicit summation formula for the 2VPmTPpSn(x,y;λ;μ)holds true:

Proof.Replacing t→t+u in generating function(2.1)and using the following rule:

in the r.h.s.of the resultant equation,we find

Replacing x by z in the above equation and then equating the resultant equation to the above equation,we find

which on expanding the exponential in the r.h.s.gives

Now,using Eq.(2.21)in the r.h.s.of Eq.(2.24)and replacing n→n?l and k→k?m in the r.h.s.of the resultant equation,we find

Finally,on equating the coefficients of the same powers of t and u in Eq.(2.25),we are led to assertion(2.20).In the next section,examples of some members belonging to the 2VPmTPpSn(x,y;λ;μ)are considered.

3 Examples

The 2VGP family pn(x,y)which are classified as an important general class of special functions due to wide range of applications contains a number of important special polynomials of two variables.Certain members belonging to the 2VGP family pn(x,y)are considered in Section 1.We note that corresponding to each member belonging to the 2VGP pn(x,y),there exists a new special polynomial belonging to the 2VPmTPpSn(x,y;λ;μ)family.Thus,by making suitable choice for the function φ(y,t)in Eq.(2.1),we get the generating function for the corresponding member belonging to the 2VPmTPpSn(x,y;λ;μ)family.The other properties of these special polynomials can be obtained from the results derived in the previous section.

We consider the following examples:

Example 3.1.Taking φ(y,t)=eytm(for which the 2VGP pn(x,y)reduce to the GHP(x,y)Table 1(I))in the l.h.s.of generating function(2.1),we find that the resultant Gould-Hopper Peter polynomials(GHPP),denoted byH(m)Sn(x,y;λ;μ)in the r.h.s.are defined by the following generating function:

The series definitions and other results for the GHPPH(m)Sn(x,y;λ;μ)are given in Table 3.

Remark 3.1.Since for m=2,the GHP(x,y)reduce to the 2VHKdFP Hn(x,y)(Table 1(II)).Therefore,taking m=2 in Eq.(3.1),we get the following generating function for the 2-variable Hermite Peters polynomials(2VHPP),denoted byHSn(x,y;λ;μ):

The series definitions and other results for the 2VHPPHSn(x,y;λ;μ)can be obtained by taking m=2 in the results given in Table 3.

Remark 3.2.Since for x→2x and y=?1 the 2VHKdFP Hn(x,y)reduce to the classical Hermite polynomials Hn(x)[1].Therefore,taking x→2x and y=?1 in Eq.(3.2),we get the following generating function for the Hermite Peters polynomials(HPP),denoted by

The series definitions and other results for the HPPHSn(x;λ;μ)can be obtained by taking m=2,x→2x and y=?1 in the results given in Table 3.

Taking suitable values of the parameters in the results of the GHPPH(m)Sn(x,y;λ;μ)and in view of Eqs.(1.15a)and(1.15b),we can find the corresponding results for the mixed special polynomials related toH(m)Sn(x,y;λ;μ).We use the suitable notations forthese polynomials and present their generating functions and series definitions in Table 4.

Table 3:Results forH(m)Sn(x,y;λ;μ).

Table 4:Special cases of the GHPPH(m)Sn(x,y;λ;μ).

We note that for m=2 the results derived above for the GHBlPH(m)Bln(x,y;λ)and GHChPH(m)Chn(x,y)give the corresponding results for the 2-variable Hermite Boole polynomials(2VHBlP)HBln(x,y;λ)and 2-variable Hermite Changee polynomials(2VHChP)HChn(x,y),respectively.Again,for m=2,x→2x and y=?1,we get the corresponding results for the Hermite Boole polynomials(HBlP)HBln(x;λ)and Hermite Changee polynomials(HChP)HChn(x).Also in the results of the mixed special polynomials with(x,y)as base,we obtain the results(with corresponding changes in values of indices and variable)for the corresponding mixed special polynomials with Hn(x,y)and Hn(x)as base.

Example 3.2.Taking φ(y,t)=C0(?ytm)(for which the 2VGP pn(x,y)reduce to the 2VGLPmLn(y,x)Table 1(III))in the l.h.s.of generating function(2.1),we find that the resultant 2-variable generalized Laguerre Peters polynomials(2VGLPP),denoted bymLSn(y,x;λ;μ)in the r.h.s.are defined by the following generating function:

The series definitions and other results for the 2VGLPPmLSn(y,x;λ;μ)are given in Table 5.

Remark 3.3.Since for m=1 and y→?y,the 2VGLPmLn(y,x)reduce to the 2VLP Ln(y,x)(Table 1(IV)).Therefore,taking m=1 and y→?y in Eq.(3.4),we get the following generating function for the 2-variable Laguerre Peters polynomials(2VLPP),denoted byLSn(y,x;λ;μ)

The series definitions and other results for the 2VLPPLSn(y,x;λ;μ)can be obtained by taking m=1 and y→?y in the results given in Table 5.

Table 5:Results formLSn(y,x;λ;μ).

Remark 3.4.Since for x=1,the 2VLP Ln(y,x)reduce to the classical Laguerre polynomials Ln(y)[1].Therefore,taking x=1 in Eq.(3.5),we get the following generating function for the Laguerre Peters polynomials(LPP),denoted byLSn(y;λ;μ)

The series definitions and other results for the LPPLSn(y;λ;μ)can be obtained by taking m=1,y→?y and x=1 in the results given in Table 5.

Taking suitable values of the parameters in the results of the 2VGLPPmLSn(y,x;λ;μ)and in view of Eqs.(1.15a)and(1.15b),we can find the corresponding results for the mixed special polynomials related tomLSn(y,x;λ;μ).We use the suitable notations for these polynomials and present their generating functions and series definitions in Table 6.

We note that for m=1,y→?y the results derived above for the 2VGLBlPmLBln(y,x;λ)and 2VGLChPmLChn(y,x)give the corresponding results for the 2-variable Laguerre Boole polynomials(2VLBlP)LBln(x,y;λ)and 2-variable Laguerre Changee polynomials(2VLChP)LChn(x,y),respectively.Again,for m=1,y→?y and x=1,we get the corresponding results for the Laguerre Boole polynomials(LBlP)LBln(x;λ)and Laguerre Changee polynomials(LChP)LChn(x).Thus,in the results of the mixed special polynomials withmLn(y,x)as base,we obtain the results(with corresponding changes in values of parameters)for the corresponding mixed special polynomials with Ln(y,x)and Ln(y)as base.

Example 3.3.By taking φ(y,t)=(for which the 2VGP pn(x,y)reduce to the 2VTEP(x,y)Table 1(V))in the l.h.s.of generating function(2.1),we find that the resultant 2-variable truncated exponential Peters polynomials(2VTEPP),denoted bye(r)Sn(x,y;λ;μ)in the r.h.s.are defined by the following generating function:

The series definitions and other results for the 2VTEPPe(r)Sn(x,y;λ;μ)are given in Table 7.

Remark 3.5.Since for r=2,the 2VTEP e(r)(x,y)of order r reduce to the 2VTEP[2]en(x,y)(Table 1(VI)).Therefore,taking r=2 in Eq.(3.7),we get the following generating func-tion for the 2-variable truncated exponential Peters polynomials(2VTEPP),denoted by[2]eSn(x,y;λ;μ):

Table 6:Special cases of the 2VGLPPmLSn(y,x;λ;μ).

Table 7:Results fore(r)Sn(x,y;λ;μ).

The series definitions and other results for the 2VTEPP[2]eSn(x,y;λ;μ)can be obtained by taking r=2 in the results given in Table 7.

Remark3.6.Sincefor y=1,the[2]en(x,y)reduce to the truncatedex ponential polynomials[2]en(x)[6].Therefore,taking y=1 in Eq.(3.8),we get the following generating function for the truncated exponential Peters polynomials(TEPP),denoted by[2]eSn(x;λ;μ):

The series definitions and other results for the TEPP[2]eSn(x;λ;μ)can be obtained by taking r=2 and y=1 in the results given in Table 7.

Taking suitable values of the parameters in the results of the 2VTEPPe(r)Sn(x,y;λ;μ)and in view of Eqs.(1.15a)and(1.15b),we can find the corresponding results for the mixed special polynomials related toe(r)Sn(x,y;λ;μ).We use the suitable notations for these polynomials and present their generating functions and series definitions in Table 8.

It is also important to observe that taking r=2 in the results of the mixed special polynomials with(x,y)as base,we obtain the results for the corresponding mixed special polynomials with[2]en(x,y)as base.Also,taking r=2 and y=1,we obtain the results for the mixed special polynomials with[2]en(x)as base.

Further,we note that the multiplicative and derivative operators,differential equations,operational rule and summation formulae for the polynomials mentioned in Table??,Table 6 and Table 8 can also be obtained by taking suitable values of the parameters in the corresponding results of the GHPPH(m)Sn(x,y;λ;μ),2VGLPPmLSn(x,y;λ;μ)and 2VTEPPe(r)Sn(x,y;λ;μ),respectively.

Table 8:Special cases of the 2VTEPPe(r)Sn(x,y;λ;μ).

In the next section,numbers related to some mixed special polynomials are explored.

4 Concluding remarks

We know that the Peters,Boole and changee numbers have deep connections with number theory and occur in combinatorics.These numbers appear as special values of the Peters,Boole and changee polynomials as indicated in Eq.(1.20).

The Peter numbers Sn(λ;μ)are defined by the generating function(1.18a).In view of relations(1.15a)and(1.15b),we find the following special cases of Sn(λ;μ):

Here,we explore the numbers corresponding to the HPPHSn(x;λ;μ)defined by the generating function(3.3).Taking m=2,y=?1 and replacing x by 2x in series definition of the GHPPH(m)Sn(x,y;λ;μ)(Table 3(I)),we find the following series definition of the HPPHSn(x;λ;μ)

Since,the HPPHSn(x;λ;μ)are defined in terms of the Hermite polynomials.Therefore,in order to find the Hermite Peters numbers,denoted byHSn(λ;μ),we require the Hermite numbers.

We recall that the Hermite numbers Hnare the values of the Hermite polynomials Hn(x)at zero argument,that is

From the generating function of the Hermite polynomials

it follows that

A closed formula for Hnis given as:

Now,taking x=0 in both sides of definition(4.2)and using the notation

in the l.h.s.and notation(4.3)in the r.h.s.of the resultant equation,we find that the Hermite Peters numbersHSn(λ;μ)are defined as:

Takingμ=1 in Eq.(4.8)and using relation(4.1a)in the r.h.s.and denoting the resultant Hermite Boole numbers in the l.h.s.byHBln(λ),that is

we find the following series definition of theHBln(λ):

Next,taking λ=1 and μ=1 in Eq.(4.8)and using relation(4.1b)in the r.h.s.and denoting the resultant Hermite Changee numbers,in the l.h.s.byHChn,that is

we find the following series definition of theHCh:

In our next investigation,we derive certain results for the 2-variable Apostol type and related polynomials by means of umbral technique.

Acknowledgements

This work has been done under UGC-BSR Reaserch Start-Up-Grant(Office Memo No.30-90/2015(BSR))awarded to the author by the University Grants Commission(UGC),Government of India,New Delhi.