The present study deals with some new properties for the Gottlieb polynomials in one and several variables

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Vol. 19 (2018), No. 2, pp. 835–845 DOI: 10.18514/MMN.2018.2188

SOME NEW PROPERTIES OF UNIVARIATE AND MULTIVARIATE GOTTLIEB POLYNOMIALS

ESRA ERKUS¸-DUMAN Received 21 December, 2016

Abstract. The present study deals with some new properties for the Gottlieb polynomials in one and several variables. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties and also some special cases for these polynomials.

2010Mathematics Subject Classification: 33C45

Keywords: Gottlieb polynomials, generating function, Pochhammer symbol, recurrence relation

1. INTRODUCTION

A theoretical connection with the unification of generating functions has great importance in the study of special functions. Steps forward in this directions has been made by some researchers [1,2,7,8].

The (univariate) Gottlieb polynomials were introduced and investigated in 1938, and then have been cited in several articles. These polynomials are defined by [10]

'n.xI/De ⁿ

n

X

kD0

n k

! x k

!

1 ek

(1.1) De ⁿ2F1

n; xI1I1 e

;

where 2F1 denotes Gauss’s hypergeometric series whose natural generalization of an arbitrary number of p numerator andq denominator parameters.p; q 2N0 WD N[ f0g/is called and denoted by the generalized hypergeometric seriespFq defined by

pFq

˛1; :::; ˛pI ˇ1; :::; ˇqI ´

D

1

X

nD0

.˛1/n::: .˛p/n

.ˇ1/n::: .ˇq/n

´ⁿ nŠ D pFq ˛1; :::; ˛pIˇ1; :::; ˇqI´

:

c 2018 Miskolc University Press

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Here./denotes the Pochhammer symbol defined (in terms of gamma function) by

./D .C/

./ .2CnZ₀/ D

1; ifD0I 2Cnf0g .C1/:::.Cn 1/; ifDn2NI 2C

and Z₀ denotes the set of nonpositive integers and ./ is the familiar Gamma function. Furthermore, the hypergeometric function2F1 has the following integral representation [12]:

2F1.a; bIcI´/D .c/

.b/ .c b/

1

Z

0

t^{b 1}.1 t /^{c b 1}.1 ´t / ^adt; (1.2) where Re.c/ >Re.b/ > 0, andjarg.1 ´/j< .

Gottlieb [10] presented following generating functions for his polynomials'n.xI/ (see also [12, p. 303]), which is denoted byln.x/in [10],

1

X

nD0

'n.xI/ tⁿD.1 t /^x

1 t e x 1

.jtj< 1/ ;

1

X

nD0

./_n

nŠ 'n.xI/ tⁿD

1 t e

2F1

2 4

; xI 1I

1 e

t 1 t e

3 5;

1

X

nD0

'n.xI/tⁿ

nŠ De^t 1F1

h

xC1I1I

1 e

ti :

In addition to this, we can easily see the following relation for the Gottlieb polynomials [14, p. 449, 20 (i)]:

1

X

nD0

kCn n

!

'_k_C_n.xI/ tⁿD.1 t /^{x k}

1 t e x 1

'_k xIln e t 1 t

!!

: (1.3) Recently, Khan and Akhlaq [11] introduced and investigated Gottlieb polynomials in two and three variables to give their generating functions. Choi [5], by modify- ing Khan and Akhlaq’s method, showed how to generalize the Gottlieb polynomials in several variables to present two generating functions of the generalized Gottlieb polynomials. Furthermore, he derives q-extension of a generalization of Gottlieb polynomials in three variables and gives some formulas deducible from a generalization of these polynomials in several variables (see [4]). Choi introduced a several variable analogue of the Gottlieb polynomials as follows.

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Definition 1. An extension of the Gottlieb polynomials'n.xI/inmvariables is defined by [5]

'_n^m.x1; :::; xmI1; :::; m/Dexp. nm/

n

X

r1D0 n r1

X

r2D0

n r1 r2

X

r3D0

:::

n r1 r2 ::: rm 1

X

rmD0

. n/_ı_m

m

Q

jD1

xj

r_j

Q

jD1

1 e^j

rj

m

Q

jD1

rjŠımŠ

; (1.4)

wheren; m2Nand, for convenience, mWD

m

X

jD1

j and ımWD

m

X

jD1

rj: (1.5)

It is noted that the special casemD1of (1.4) reduces immediately to the Gottlieb polynomials in (1.1). Multivariable Gottlieb polynomials defined by (1.4) have the following two generating functions [5]:

1

X

nD0

'_n^m.x1; :::; xmI1; :::; m/ tⁿ

D 1 t e ^m

m

P

jD1

x_j

! 1 mY

jD1

1 t e^j ^mx_j

; (1.6)

wherem2Nandmis given in (1.5).

1

X

nD0

./_n'_n^m.x1; :::; xmI1; :::; m/tⁿ nŠ D 1 t e ^m

F_D^.m/Œ; x1; :::; xmI1I t

e¹ 1 e ^m 1 t e ^m ; :::;

t

e^m 1 e ^m 1 t e ^m

3 5;

whereF_D^.m/Œ:denotes one of the Lauricella series inmvariables [13, p.33, Eq.(4)]

defined by

F_D^.m/Œa; b1; :::; bmIcIx1; :::; xmD

1

X

r1;r2;:::;rmD0

.a/_ı_m.b1/_r₁::: .bm/_r_m .c/_ı_m

x₁^r¹ r1Š:::x^rm^m

rmŠ .maxfjx1j; :::;jxmjg< 1/

andm; ımare given in (1.5).

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The main object of this paper is to study different properties of Gottlieb polynomials in one and several variables. Various families of multilinear and multilateral generating functions are derived for these polynomials. Other miscellaneous properties of these multivariable polynomials are also discussed. Some special cases of the results presented in this study are also indicated.

2. GENERATING FUNCTIONS

In this section, firstly, we prove a theorem which gives bilateral generating functions relations for the Gottlieb polynomials defined by (1.1).

Theorem 1. Let

N_n;m;q^;p .y1; :::; ysI´/WD

Œn=q

X

kD0

mCn n qk

!

a_k˝_C_pk.y1; :::; ys/´^k: (2.1) If

m;qŒxIIy1; :::; ysIt D

1

X

nD0

an'mCq n.xI/ ˝Cpn.y1; :::; ys/tⁿ; then, for every nonnegative integerm, we have

1

X

nD0

'mCn.xI/ N_n;m;q^;p .y1; :::; ysI´/ tⁿ (2.2) D.1 t /^{x m}

1 t e x 1

m;q

"

xIln e t 1 t

!

Iy1; :::; ysI´ t

1 t q#

: Proof. If we denote the left-hand side of.2:2/byT and use.2:1/;then we obtain

T D

1

X

nD0

'mCn.xI/

Œn=q

X

kD0

mCn n qk

!

a_k˝_C_pk.y1; :::; ys/´^ktⁿ: ReplacingnbynCqkand then using relation.1:3/we may write

T D

1

X

nD0 1

X

kD0

'mCnCqk.xI/ mCnCqk n

!

ak˝Cpk.y1; :::; ys/´^ktⁿ^C^qk

D

1

X

kD0

.1 t /^{x m qk}

1 t e x 1

'_m_C_qk xIln e t 1 t

!!

a_k ˝_C_pk.y1; :::; ys/ ´t^qk

D.1 t /^{x m}

1 t e x 1

m;q

"

xIln e t 1 t

!

Iy1; :::; ysI´ t

1 t q#

;

which completes the proof.

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Now we give a theorem which is about several families of bilinear and bilateral generating functions for the multivariable Gottlieb polynomials

'_n^m.x1; :::; xmI1; :::; m/.

Theorem 2. Corresponding to an identically non-vanishing function

˝.y1; :::; ys/ofscomplex variablesy1; :::; ys.s2N/and of complex order, let ;.y1; :::; ysI´/WD

1

X

kD0

a_k˝_C_k.y1; :::; ys/´^k; wherea_k ¤0 ; ; 2Cand

_n;m;p^; .x1; :::; xmI1; :::; mIy1; :::; ysI / (2.3) W D

Œn=p

X

kD0

a_k'_{n pk}^m .x1; :::; xmI1; :::; m/ ˝_C_k.y1; :::; ys/^k .n; p2N/:Then we have

1

X

nD0

_n;m;p^;

x1; :::; xmI1; :::; mIy1; :::; ysI t^p

tⁿ (2.4)

D 1 t e ^m

m

P

jD1

xj

! 1 mY

jD1

1 t e^j ^mx_j

;.y1; :::; ysI/ provided that each member of.2:4/exists.

Proof. If we denote the left-hand side of.2:4/byS and use.2:3/;then we obtain SD

1

X

nD0 Œn=p

X

kD0

a_k'_{n pk}^m .x1; :::; xmI1; :::; m/ ˝_C_k.y1; :::; ys/^kt^{n pk}: ReplacingnbynCpkwe may write from the generating function (1.6) that

SD

1

X

nD0 1

X

kD0

ak'_n^m.x1; :::; xmI1; :::; m/ ˝Ck.y1; :::; ys/^ktⁿ

D

1

X

nD0

'_n^m.x1; :::; xmI1; :::; m/ tⁿ

1

X

kD0

a_k˝_C_k.y1; :::; ys/^k

D 1 t e ^m

m

P

jD1

x_j

! 1 mY

jD1

1 t e^j ^mx_j

;.y1; :::; ysI/;

which completes the proof.

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It is possible to give many applications of our theorems with help of appropriate choices of the multivariable functions˝_C_k.y1; :::; ys/; k2N0; s2N.

For example, set sDm and take ˝_C_k.y1; :::; ym/Dh^.ˇ_C¹^;:::;ˇ_k ^m^/.y1; :::; ym/ in Theorem 2. Recall that, by h^.˛n¹^;:::;˛^r^/.x1; :::; xr/; we denote the multivariable Lagrange-Hermite polynomials (see, e.g., [2]) generated by

r

Y

jD1

n

.1 xjt^j/ ^˛^jo D

1

X

nD0

h^.˛_n¹^;:::;˛^r^/.x1; :::; xr/tⁿ; (2.5) where jtj<min

n

jx1j ¹;jx2j ¹⁼²; :::;jxrj ^1=ro

. We also know that h^.˛_n¹^{; : : : ;˛}^r^/.x1; :::; xr/D X

k₁C2k₂C: : :CrkrDn

.˛1/k₁: : : .˛r/k_r

x₁^k¹ k1Š: : :x_r^k^r

krŠ: Then, from Theorem2, we obtain the following result which is a class of bilateral generating functions for the multivariable Lagrange-Hermite polynomials and the multivariable Gottlieb polynomials.

Corollary 1. If

;.y1; :::; ymI´/WD

1

X

kD0

a_kh^.ˇ_C¹^;:::;ˇ_k ^m^/.y1; :::; ym/´^k; .a_k¤0; ; 2C/ ; and

_n;m;p^; .x1; :::; xmI1; :::; mIy1; :::; ymI / W D

Œn=p

X

kD0

a_k'_{n pk}^m .x1; :::; xmI1; :::; m/ h^.ˇ_C¹^;:::;ˇ_k ^m^/.y1; :::; ym/^k; wheren; p2N, then we have

1

X

nD0

_n;m;p^;

x1; :::; xmI1; :::; mIy1; :::; ymI t^p

tⁿ (2.6)

D 1 t e ^m

m

P

jD1

x_j

! 1 mY

jD1

1 t e^j ^mxj

;.y1; :::; ymI/ provided that each member of.2:6/exists.

Remark 1. Using the generating relation (2.5) for the multivariable Lagrange- Hermite polynomials and takingak D1; D0; D1;we have

1

X

nD0 Œn=p

X

kD0

'_{n pk}^m .x1; :::; xmI1; :::; m/ h^.ˇ_k ¹^;:::;ˇ^m^/.y1; :::; ym/^kt^{n pk}

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D 1 t e ^m

m

P

jD1

xj

! 1 mY

jD1

1 t e^j ^mxj

.1 y_j^j/ ^ˇ^j;

wheremWD

m

P

jD1

j andjj<min n

jy1j ¹;jy2j ¹⁼²; :::;jymj ^1=mo :

Also, choosing sDm and˝_C_k.y1; :::; ym/D'^m_C_k.y1; :::; ymI1; :::; m/ ;

; 2N0;in Theorem2, we obtain the following class of bilinear generating functions for the multivariable Gottlieb polynomials defined by (1.4).

Corollary 2. If

;.y1; :::; ymI´/WD

1

X

kD0

a_k'^m_C_k.y1; :::; ymI1; :::; m/ ´^k; .a_k ¤0; ; 2C/ ; and

_n;m;p^; .x1; :::; xmI1; :::; mIy1; :::; ymI / W D

Œn=p

X

kD0

a_k'_{n pk}^m .x1; :::; xmI1; :::; m/ '^m_C_k.y1; :::; ymI1; :::; m/ ^k; wheren; p2N, then we have

1

X

nD0

_n;m;p^;

x1; :::; xmI1; :::; mIy1; :::; ymI t^p

tⁿ (2.7)

D 1 t e ^m

m

P

jD1

x_j

! 1 mY

jD1

1 t e^j ^mxj

;.y1; :::; ymI/ provided that each member of.2:7/exists.

Remark2. Using the generating relation (1.6) for the multivariable Gottlieb polynomials and takinga_k D1; D0; D1;we have

1

X

nD0 Œn=p

X

kD0

'_{n pk}^m .x1; :::; xmI1; :::; m/ '_k^m.y1; :::; ymI1; :::; m/ ^kt^{n pk}

D 1 t e ^m

m

P

jD1.^xjCyj/

! 2 mY

jD1

1 t e^j ^m xj

1 e^j ^m yj

;

where mWD

m

P

jD1

j:

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Furthermore, for every suitable choice of the coefficientsak .k2N0/;if the multivariable function˝_C_k.y1; :::; ys/ .s2N/is expressed as an appropriate product of several simpler functions, the assertions of Theorems1 and2can be applied in order to derive various families of multilinear and multilateral generating functions for the univariate and multivariate Gottlieb polynomials.

3. MISCELLANEOUS PROPERTIES

In this section, firstly, we obtain a recurrence relation for the multivariable Gottlieb polynomials. Then, using the relation between Gottlieb and Jacobi polynomials, we derive some properties for the Gottlieb polynomials.

Theorem 3. Multivariable Gottlieb polynomials have the following recurrence relation:

.nC1/ '_n^m_C₁Ceⁱ ²^mn'_{n 1}^m De ^m 0

@1CnCneⁱC

m

X

jD1

xj

1 e^j 1 A'_n^m; . iD1; :::; m I n2/ :

Proof. By differentiating each member of the generating function relation (1.6) with respect to t and making some necessary adjustments and then identifying the corresponding coefficients oftⁿ, we arrive at the desired recurrence relation.

It is known that

'n.xI/DP_n^{.0;x n/}

2e 1

; (3.1)

where the classical Jacobi polynomialsPn^{.˛;ˇ /}.´/are defined by [12]

P_n^{.˛;ˇ /}.´/D

n

X

kD0

nC˛ k

! nCˇ

n k

!xC1 2

kx 1 2

n k

:

Equation (3.1) can be used to deduce numerous properties and characteristics of the Gottlieb polynomials from those of the Jacobi polynomials.

In 1966, Izuru Fujiwara studied extended Jacobi polynomials F_n^{.˛;ˇ /}.xIa; b; c/

which are defined by the Rodrigues formula [9]

F_n^{.˛;ˇ /}.xIa; b; c/D. c/ⁿ

nŠ .x a/ ^˛.b x/ ^ˇ dⁿ

dxⁿ n

.x a/ⁿ^C^˛.b x/ⁿ^C^ˇo

.c > 0/ :

Szeg¨o [15] showed thatFn^{.˛;ˇ /}.xIa; b; c/polynomials are a constant multiple of the classical Jacobi polynomialsP_n^{.˛;ˇ /}.x/in the form

F_n^{.˛;ˇ /}.xIa; b; c/D fc .a b/gⁿP_n^{.˛;ˇ /}

2 .x a/

a b C1

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or, equivalently,

P_n^{.˛;ˇ /}.x/D fc .a b/g ⁿF_n^{.˛;ˇ /} 1

2faCbC.a b/ xg Ia; b; c

: (3.2)

In addition to this, we know that the following equality [3]:

g_n^{.˛;ˇ /}.x; y/D.y x/ⁿP. ˛ n; ˇ n/

n

xCy

x y

; (3.3)

whereg_n^{.˛;ˇ /}.x; y/are known as the Lagrange polynomials generated by [6]

.1 xt / ^˛.1 yt / ^ˇ D

1

X

nD0

g_n^{.˛;ˇ /}.x; y/ tⁿ

jtj<min n

jxj ¹;jyj ¹o : Theorem 4. For the Gottlieb polynomials'n.xI/, we have .i / 'n.xI/D fc .a b/g ⁿFn^{.0;x n/}

1 2

n

aCbC.a b/

2e 1

o Ia; b; c

.i i / 'n

xIln^{a b}_a

D.b a/ ⁿgn^{. n; x/}.a; b/ :

Proof. .i / Using relations (3.1) and (3.2), we arrived the result.

.i i / Using relations (3.1) and (3.3), we get the desired result.

Theorem 5. Gottlieb polynomials have the following integral representation:

'n.xI/D sinx

eⁿ

1

Z

0

t ^{x 1}.1 t /^x

1

1 e tn

dt;

where 1 < x < 0:

Proof. If we use definition of Gottlieb polynomials given by (1.1) in (1.2) and make some necessary calculation, we easily arrived this result.

Finally, using the fact that (see [14]) P_n^{.˛;ˇ /}.x/D 1

.˛CˇCnC1/

Z1

0

t^˛^C^ˇ^Cⁿe ^tL^.˛/_n

.1 x/ t 2

dt;

whereL^.˛/n .x/are the Laguerre polynomials given by the Rodrigues formula [12]

L^.˛/_n .x/D x ^˛e^x nŠ

dⁿ

dxⁿ e ^xxⁿ^C^˛

;

and also considering (3.1), we immediately obtain the following result.

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Theorem 6. Gottlieb polynomials have the following integral representation:

'n.xI/D 1 x .x/

Z1

0

t^xe ^tLn

1 e

t dt:

ACKNOWLEDGEMENT

The author would like to thank the referees for carefully reading the manuscript.

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Author’s address

Esra Erkus¸-Duman

Gazi University, Department of Mathematics, Teknikokullar TR-06500, Ankara, Turkey E-mail address:eduman@gazi.edu.tr; eerkusduman@gmail.com