Open access peer-reviewed chapter

Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions of the Form F(t)xG(t)α

By Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya

Submitted: July 28th 2018Reviewed: November 2nd 2018Published: April 4th 2019

DOI: 10.5772/intechopen.82370

Downloaded: 148

Abstract

In this chapter, we study properties of polynomials defined by generating functions of the form A t x α = F t x ⋅ G t α . Based on the Lagrange inversion theorem and the theorem of logarithmic derivative for generating functions, we obtain new properties related to the compositional inverse generating functions of those polynomials. Also we study the composition of generating functions R tA t , where A t is the generating function of the form F t x ⋅ G t α . We apply those results for obtaining explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials.

Keywords

  • polynomial
  • identity
  • generating function
  • composita
  • composition
  • compositional inverse

1. Introduction

Generating functions are a powerful tool for solving problems in number theory, combinatorics, algebra, probability theory, and other fields of mathematics. One of the advantages of generating functions is that an infinite number sequence can be represented in a form of a single expression. Many authors have studied generating functions and their properties and found applications for them (for instance, Comtet [1], Flajolet and Sedgewick [2], Graham et al. [3], Robert [4], Stanley [5], and Wilf [6]).

Generating functions have an important role in the study of polynomials. Vast investigations related to the generating functions for many polynomials can be found in many books and articles (e.g., see [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]).

A special place in this area is occupied by research in the field of obtaining new identities for polynomials and special numbers with using their generating functions. Interesting results in the field of obtaining new identities for polynomials can be found in some recent works by Simsek [18, 19, 20], Kim et al. [21, 22], and Ryoo [23, 24, 25].

Another trend in study of polynomials is getting new representation and explicit formulas for those polynomials. For instance, Qi has recently established explicit formulas for the generalized Motzkin numbers in [26] and the central Delannoy numbers in [27]. One can find interesting results in papers of Srivastava [28, 29], Cenkci [30], and Boyadzhiev [31].

In this chapter, we obtain some interesting properties of polynomials defined by generating functions of the form FtxGtα. As an application, we give some new identities for the Bernoulli, Euler, Frobenius-Euler, Sylvester, Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials.

According to Stanley [32], ordinary generating functions are defined as follows:

Definition 1. An ordinary generating function of the sequence ann0is the formal power series

Ax=a0+a1x+a2x2+=n0anxn.E1

Kruchinin et al. [33, 34, 35] introduced the mathematical notion of the composita of a given generating function, which can be used for calculating the coefficients of a composition of generating functions.

Definition 2. The composita of the generating function Fx=n>0fnxnis the function with two variables

FΔnk=πkCnfλ1fλ2fλk,E2

where Cnis the set of all compositions of an integer nand πkis the composition ninto kparts such that i=1kλi=n.

Using the expression of the composita of a given generating function FΔnk, we can get powers of the generating function Fx:

Fxk=nkFΔnkxn.E3

Compositae also can be used for calculating the coefficients of generating functions obtained by addition, multiplication, composition, reciprocation, and compositional inversion of generating functions (for details see [33, 34, 35]).

By the reciprocal generating function we mean the following [6]:

Definition 1. A reciprocal generating function Axof a generating function Bx=n0bnxnis a power series such that satisfies the following condition:

AxBx=1.E4

By the compositional inverse generating function we mean the following:

Definition 2. A compositional inverse Fx¯of generating function Fx=n>0fnxnwith f10is a power series such that satisfies the following condition:

FFx¯=x.E5

Also the compositional inverse can be written as F1xor Fx¯=RevF.

For example, we will use the following formulas:

If we consider the composition Ax=RFx=n0anxnof generating functions Rx=n0rnxnand Fx=n>0fnxn, then we can get the values of the coefficients anby using the following formula ([35], Eq. (17)):

an=r0,forn=0;k=1nFΔnkrk,otherwise.E6

If we consider the composition Ax=RFx=n>0anxnof generating functions Rx=n>0rnxnand Fx=n>0fnxn, then we can get the values of the composita AΔnkby using the following formula ([35]):

AΔnk=m=knFΔnmRΔmk.E7

2. Main results

Let us consider a special case of generating functions that can be presented as the product of the powers of generating functions FtxGtα. For such generating functions, we obtain several properties, which are given in the following theorem:

Theorem 1. If Atis a generating function of the following form:

At=FtxGtα=n0Anxαtn,E8
then:
  1. For the composition of generating functions Dt=CBt=n0Bntn, where Bt=tAtand Ct=n0Cntn, we have

    Dn=Dnxα=k=1nAnkkxCk,D0=C0;E9

  2. For the compositional inverse generating function B¯tof Bt=tAt, we have

    B¯t=n>01nAn1nxtn;E10

  3. We have the following identities

m=knAnmmxkmAmkmx=δn,kE11

and

m=knmnAnmnxAmkkx=δn,k,E12

where δn,kis the Kronecker delta.

Proof. First we get the k-th power of the generating function Bt=tAt

Btk=tAtk=tkFtxkGtαk==tkn0Ankxtn=nkAnkkxtn.

Hence, the composita of Bt=tAtis

BΔnk=Ankkx.E13

Using Eqs. (6) and (13), we get Eq. (9).

According to [36], the composita of the compositional inverse generating function A¯tof At=n>0antnis

A¯Δnk=knRΔ2nkn,E14

where RΔnkis the composita of the generating function Rt=t2At.

For getting the composita of the compositional inverse generating function B¯tof Bt=tAt, we need to know the composita of the generating function

Rt=t2Bt=t2tAt=tAt.E15

Then we get the k-th power of the generating function Rt=tAt

Rtk=tAtk=tkFtxkGtαk==tkn0Ankxtn=nkAnkkxtn.E16

Hence, the composita of Eq. (15) is

RΔnk=Ankkx.E17

Using Eqs. (14) and (17), we get

B¯Δnk=knRΔ2nkn=knA2nknnx=knAnknx.E18

For k=1, we get Eq. (10).

Applying Eq. (7) for the composition Ct=BB¯t=t, we get

CΔnk=m=knB¯ΔnmBΔmk==m=knmnAnmnxAmkkx=δn,k.E19

Applying Eq. (7) for the composition Dt=B¯Bt=x, we get

DΔnk=m=knBΔnmB¯Δmk==m=knAnmmxkmAmkmx=δn,k.E20

As an application of Theorem 1, we present several examples of its usage for such polynomials as the Bernoulli, Euler, Frobenius-Euler, Sylvester, Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner.

2.1 Generalized Bernoulli polynomials

The generalized Bernoulli polynomials are defined by the following generating function [37, 38]:

Btxα=exttet1α=etxtet1α=n0Bnαxtnn!,E21

where

Bnαx=i=0nn!n+i!n+αnii+α1ij=0i1jijx+jn+i.E22

According to Eq. (13), the composita for the generating function Dt=tBtxαis

DΔnk=Bnkkxnk!.E23

The triangular form of this composita is

12xα2112x212αx+3α2α242xα18x312αx2+6α22αxα3+α24824x224αx+6a2α126x3α21

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tBtxαis

D¯Δnk=knBnknxnk!.E24

The triangular form of this composita is

12x+α2136x236αx+9a2+α242x+α132x3+48αx224α2+2αx+4α3+α21248x248αx+12α2+α126x+3α21

Also we can get the following new identities for the generalized Bernoulli polynomials:

m=knmnBnmnxnm!Bmkkxmk!=δn,kE25
and

m=knBnmmxnm!kmBmkmxmk!=δn,k.E26

2.2 Generalized Euler polynomials

The generalized Euler polynomials are defined by the following generating function [37]:

Etxα=ext2et+1α=etx2et+1α=n0Enαxtnn!,E27

where

Enαx=i=0n12ii+α1ij=0i1jijx+jn.E28

According to Eq. (13), the composita for the generating function Dt=tEtxαis

DΔnk=Enkkxnk!.E29

The triangular form of this composita is

12xα214x24αx+α2α82xα18x312αx2+6α26αxα3+3α2488x28αx+2a2α46x3α21

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tEtxαis

D¯Δnk=knEnknxnk!.E30

The triangular form of this composita is

12x+α2112x212αx+3a2+α82x+α132x3+48αx224α2+6αx+4α3+3α21216x216αx+4α2+α46x+3α21

Also we can get the following new identities for the generalized Euler polynomials:

m=knmnEnmnxnm!Emkkxmk!=δn,kE31
and

m=knEnmmxnm!kmEmkmxmk!=δn,k.E32

2.3 Frobenius-Euler polynomials

The Frobenius-Euler polynomials are defined by the following generating function [39]:

Htxαλ=ext1λetλα=etx1λetλα=n0Hnαxλtnn!,E33

where

Hnαxλ=i=0n11λii+α1ij=0i1jijx+jn.E34

According to Eq. (13), the composita for the generating function Dt=tHtxαλis

DΔnk=Hnkkxλnk!.E35

The triangular form of this composita is

1λ1x+αλ11λ22λ+1x2+2λ2αx+α2+λα2λ24λ+22λ2x+2αλ11

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tHtxαλis

D¯Δnk=knHnknxλnk!.E36

The triangular form of this composita is

1λ1x+αλ113λ26λ+3x2+6λ6αx+3α2λα2λ24λ+22λ2x+2αλ11

Also we can get the following new identities for the Frobenius-Euler polynomials:

m=knmnHnmnxλnm!Hmkkxλmk!=δn,kE37
and

m=knHnmmxλnm!kmHmkmxλmk!=δn,k.E38

2.4 Generalized Sylvester polynomials

The generalized Sylvester polynomials are defined by the following generating function [40]:

Ftxα=1txeαxt=eαt1tx=n0Fnxαtn,E39

where

Fnxα=i=0nαxnini!i+x1i.E40

According to Eq. (13), the composita for the generating function Dt=tFtxαis

DΔnk=Fnkkxα.E41

The triangular form of this composita is

1α+1x1α2+2α+1x2+x22α+2x1α3+3α2+3α+1x3+3α+3x2+2x62α2+4α+2x2+x3α+3x1

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tFtxαis

D¯Δnk=knFnknxα.E42

The triangular form of this composita is

1α+1x13α2+6α+3x2x22α+2x18α3+24α2+24α+8x36α+6x2+x34α2+8α+4x2x3α+3x1

Also we can get the following new identities for the generalized Sylvester polynomials:

m=knmnFnmnxαFmkkxα=δn,kE43
and

m=knFnmmxαkmFmkmxα=δn,k.E44

2.5 Generalized Laguerre polynomials

The generalized Laguerre polynomials are defined by the following generating function [8]:

Ltxα=1tα1extt1=ett1x11tα+1=n0Lnαxtn,E45

where

Lnαx=i=0nxii!n+αni.E46

According to Eq. (13), the composita for the generating function Dt=tLtxαis

DΔnk=Lnk+k1kx.E47

The triangular form of this composita is

1x+α+11x22α+4x+α2+3α+222x+2α+21

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tLtxαis

D¯Δnk=knLnkn1nx.E48

The triangular form of this composita is

1xα113x26α+4x+3α2+5α+222x2α21

Also we can get the following new identities for the generalized Laguerre polynomials:

m=knmnLnmn1nxLmk+k1kx=δn,kE49
and

m=knLnm+m1mxkmLmkm1mx=δn,k.E50

2.6 Abel polynomials

The Abel polynomials are defined by the following generating function [8, 41]:

Atxα=eWαtxα=eWαtαx=n0Anxαtnn!,E51

where Wtis the Lambert Wfunction and

Anxα=xxαnn1.E52

According to Eq. (13), the composita for the generating function Dt=tAtxαis

DΔnk=Ankkxαnk!.E53

The triangular form of this composita is

1x1x22αx22x1x36αx2+9α2x62x22αx3x1x412αx3+48α2x264α3x244x312αx2+9α2x39x26αx24x1

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tAtxαis

D¯Δnk=knAnknxαnk!.E54

The triangular form of this composita is

1x13x2+2αx22x116x3+24αx2+9α2x64x2+2αx3x1125x4+300αx3+240α2x2+64α3x2425x3+30αx2+9α2x315x2+6αx24x1

Also we can get the following new identities for the Abel polynomials:

m=knmnAnmnxαnm!Amkkxαmk!=δn,kE55
and

m=knAnmmxαnm!kmAmkmxαmk!=δn,k.E56

2.7 Bessel polynomials

The Bessel polynomials are defined by the following generating function [8]:

Btx=ex112t=e112tx=n0Bnxtnn!,E57

where

Bnx=1,n=0;k=1n2nk1!nk!k1!xk2nk,n>0.E58

According to Eq. (13), the composita for the generating function Dt=tBtxis

DΔnk=Bnkkxnk!.E59

The triangular form of this composita is

121x2+x22x1x3+3x2+3x62x2+x3x1x4+6x3+15x2+15x244x3+6x2+3x39x2+3x24x1

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tBtxis

D¯Δnk=knBnknxnk!.E60

The triangular form of this composita is

1x13x2x22x116x312x2+3x64x2x3x1125x4150x3+75x215x2425x315x2+3x315x23x24x1

Also we can get the following new identities for the Bessel polynomials:

m=knmnBnmnxnm!Bmkkxmk!=δn,kE61
and

m=knBnmmxnm!kmBmkmxmk!=δn,k.E62

2.8 Stirling polynomials

The Stirling polynomials are defined by the following generating function [8, 42]:

Stx=t1etx=n0Snxtnn!,E63

where

Snx=i=0nx+iij=0ij!n+j!1n+jijn+jj.E64

According to Eq. (13), the composita for the generating function Dt=tStxis

DΔnk=Snkkx+k1.E65

The triangular form of this composita is

1x+1213x2+5x+224x+11x3+2x2+x486x2+11x+5123x+32115x4+30x3+5x218x857602x3+5x2+4x+1129x2+17x+882x+21

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tStxis

D¯Δnk=knSnknxn1.E66

The triangular form of this composita is

1x+2219x2+19x+1024x114x3+13x2+14x+51212x2+25x+13123x+3211875x4+8250x3+13525x2+9798x+2648576025x3+80x2+85x+302415x2+31x+1682x21

Also we can get the following new identities for the Stirling polynomials:

m=knmnSnmnxn1Smkkx+k1=δn,kE67
and

m=knSnmmx+m1kmSmkmxm1=δn,k.E68

2.9 Narumi polynomials

The Narumi polynomials are defined by the following generating function [8]:

Stxα=tln1+tα1+tx=1+txtln1+tα=n0Snxαtnn!,E69

where

Snxα=n!i=0nxnij=0ij+α1jl=0j1ljll!l+i!l+il.E70

According to Eq. (13), the composita for the generating function Dt=tStxαis

DΔnk=Snkkxnk!.E71

The triangular form of this composita is

12x+α2112x2+12α12x+3α25α242x+α18x3+12α24x2+6α222α+16x+α35α2+6α4824x2+24α12x+6α25α126x+3α21

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tStxαis

D¯Δnk=knSnknxnk!.E72

The triangular form of this composita is

12x+α2136x2+36α+12x+9α25α242xα164x396α+48x2+48α2+44α+8x+8α3+10α2+3α2448x2+48α+12x+12α2+5α126x+3α21

Also we can get the following new identities for the Narumi polynomials:

m=knmnSnmnxnm!Smkkxmk!=δn,kE73
and

m=knSnmmxnm!kmSmkmxmk!=δn,k.E74

2.10 Peters polynomials

The Peters polynomials are defined by the following generating function [8]:

Stxμλ=1+1+tλμ1+tx=1+tx11+1+tλμ=n0Snxμλtnn!,E75

where

Snxμλ=n!i=0nxnij=0i12j+μj+μ1jl=0j1ljli.E76

According to Eq. (13), the composita for the generating function Dt=tStxμλis

DΔnk=Snkkxλnk!.E77

The triangular form of this composita is

2μ2μ12xλμ22μ2μ34x24λμ+4x+λ2μ2λ2μ+2λμ22μ2xλμ23μ

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tStxμλis

D¯Δnk=knSnknxλnk!.E78

The triangular form of this composita is

2μ22μ12x+λμ22μ23μ312x2+412λμx+3λ2μ2+λ2μ2λμ23μλμ2x23μ

Also we can get the following new identities for the Peters polynomials:

m=knmnSnmnxλnm!Smkkxλmk!=δn,kE79
and

m=knSnmmxλnm!kmSmkmxλmk!=δn,k.E80

2.11 Gegenbauer polynomials

The Gegenbauer polynomials are defined by the following generating function [43]:

Ctxα=12xt+t2α=112xt+t2α=n0Cnαxtn,E81

where

Cnαx=i=0n1niinii+α1i2x2in.E82

According to Eq. (13), the composita for the generating function Dt=tCtxαis

DΔnk=Cnkx.E83

The triangular form of this composita is

12αx12α2+2αx2α4αx14α3+12α2+8αx36α2+6αx38α2+4αx22α6αx1

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tCtxαis

D¯Δnk=knCnkx.E84

The triangular form of this composita is

12αx16α22αx2+α4αx164α348α2+8αx3+24α26αx316α24αx2+2α6αx1

Also we can get the following new identities for the Gegenbauer polynomials:

m=knmnCnmxCmkx=δn,kE85

and

m=knCnmxkmCmkx=δn,k.E86

2.12 Meixner polynomials of the first kind

The Meixner polynomials of the first kind are defined by the following generating function [8, 44]:

Mtxβc=1tcx1txβ=ctc1tx11tβ=n0Mnxβctnn!,E87

where

Mnxβc=1nn!i=0nxixβnici.E88

According to Eq. (13), the composita for the generating function Dt=tMtxβcis

DΔnk=Mnkkxcnk!.E89

The triangular form of this composita is

1c1x+βcc1c22c+1x2+2β+1c22βc1x+β2+βc22c22c2x+2βcc1

Using Eq. (17), the composita for the compositional inverse generating function D¯tof Dt=tMtxβcis

D¯Δnk=knMnknxcnk!.E90

The triangular form of this composita is

11cxβcc13c26c+3x2+6β1c26βc+1x+3β2βc22c222cx2βcc1

Also we can get the following new identities for the Meixner polynomials of the first kind:

m=knmnMnmnxcnm!Mmkkxcmk!=δn,kE91

and

m=knMnmmxcnm!kmMmkmxcmk!=δn,k.E92

3. Conclusions and future developments

In this chapter, we find new explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials that are defined by generating functions of the form Atxα=FtxGtα.

A lot of studies have recently showed that polynomials are a solution for practical problems related to modeling, quantum mechanics, and other areas. So a study of obtaining explicit formulas and representations of polynomials will be important and influential. Also the further research can be conducted to find practical means of obtained properties.

Acknowledgments

This work was partially funded by the Russian Foundation for Basic Research and the government of the Tomsk region of Russian Federation (Grants No. 18-41-703006) and the Ministry of Education and Science of Russian Federation (Government Order No. 2.8172.2017/8.9, TUSUR).

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya (April 4th 2019). Obtaining Explicit Formulas and Identities for Polynomials Defined by Generating Functions of the Form <em>F</em>(<em>t</em>)<sup><em>x</em></sup> ⋅ <em>G</em>(<em>t</em>)<sup><em>α</em></sup>, Polynomials - Theory and Application, Cheon Seoung Ryoo, IntechOpen, DOI: 10.5772/intechopen.82370. Available from:

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