WIXLIB

and3(a, b, c, d; e; x, y, z) : FD,p XBp (a m n r, e a) (b)m (c)n (d)r xm y n z rB (a, e a)m! n! r!m,n,r 0ppp( x y z 1),respectively. Notice that the case p 0 gives the original functions. In obtaining thesegenerating functions we consider a new fractional derivative operator [25], namely: Z z1 pz 2 µ 1µ,pDz {f (z)} f (t) (z t)expdt,Γ ( µ) 0t(z t)(Re (µ) 0, Re (p) 0)for which p 0 gives original fractional derivative operator.In chapter 4, we present generalizations of gamma and beta functions [26] byZ p (α,β)x 1Γp (x) : tdt1 F1 α; β; t t0(Re(α) 0, Re(β) 0, Re(p) 0, Re(x) 0)andBp(α,β)Z(x, y) : 1x 1(1 t)ty 1 1 F1 α; β;0 pt(1 t) dt,(Re(p) 0, Re(x) 0, Re(y) 0, Re(α) 0, Re(β) 0)respectively. Then we use the new generalization of beta function to generalize thehypergeometric and confluent hypergeometric functions byFp(α,β)(a, b; c; z) : X(α,β)(a)nn 0Bp(b n, c b) z nB (b, c b)n!(Re(p) 0, Re(c) 0, Re(b) 0, Re(α) 0, Re(β) 0)andφ(α,β)p(b; c; z) : (α,β)XBp(b n, c b) z nB (b, c b)n 0n!,(Re(p) 0, Re(c) 0, Re(b) 0, Re(α) 0, Re(β) 0)4

respectively. The case α β gives Chaudry’s gamma, beta and hypergeometric functions. Then, we obtain some recurrence relations, transformation formulas, operationformulas and integral representations for these new generalizations.In chapter 5, we consider the following family of bivariate polynomials,Snm,N (x, y) : [ Nn ]XAm n,kk 0(n, m N0 ;xn N k y k(n N k)! k!N N),which was defined by Altın et al. [1]. We prove several general theorems involvingvarious families of generating functions for the aforementioned polynomials in theirtwo-variable notation Snm,N (x, y) by applying the method which was used recently byChen and Srivastava [12]. We also consider some applications and corollaries resultingfrom these theorems [27].5

Chapter 2PRELIMINARIES AND AUXILIARY RESULTS2.1Gamma, Beta Functions and their extended versionsIn this section, we give definitions of the gamma and beta functions and their properties. Furthermore, we give extended forms of these special functions ( see [2], [28]).Definition 2.1.1. (Euler Gamma function) For a complex number z with positive realpart (Re (z) 0), the gamma function is defined byZ Γ(z) : tz 1 exp ( t) dt.(2.1)0Using integration by parts, one can show that the following recurrance relation holdfor Γ(z):Γ(z 1) zΓ(z).This functional equation generalizes the definition n! n(n 1)! of the factorialfunction. But also, evaluating Γ (1) analytically we getΓ (1) 1.Combining these two results we see that the factorial function is a special case of thegamma function:Γ(n 1) nΓ(n) n (n 1) Γ(n 1) . n!Γ(1) n!.Another useful function is a beta function.6

Definition 2.1.2. The beta function, (also called the Eulerian integral of the first kind)is defined byZ1tx 1 (1 t)y 1 dt.B (x, y) : 0(Re (x) 0, Re (y) 0)Equivalently,Zπ2B (x, y) : 2(sin θ)2x 1 (cos θ)2y 1 dθ,0(Re (x) 0, Re (y) 0)andZ ux 1du.(1 u)x yB (x, y) : 0(Re (x) 0, Re (y) 0)The beta function is symmetric, B (x, y) B (y, x) and it is related to the gammafunction;B (x, y) Γ(x)Γ(y).Γ(x y)(2.2)(x, y 6 0, 1, 2, .)In 1994, Chaudhry and Zubair [8] introduced the following extension of gamma function.Definition 2.1.3. (see [8], [10])The Extended Gamma function is defined byZ Γp (x) : tx 1 exp t pt 1 dt,(2.3)0(Re (x) 0, Re (p) 0).In 1997, Chaudhry et al. [4] presented the following extension of Euler’s beta function.7

Definition 2.1.4. (see [4]) The Extended Beta function is defined by Z 1py 1x 1Bp (x, y) : t(1 t) exp dt,t(1 t)0(2.4)(Re(p) 0, Re(x) 0, Re(y) 0).It is clearly seem that Γ0 (x) Γ (x) and B0 (x, y) B (x, y).2.2Hypergeometric Functions and their extended versionsIn this section, we give definition and some properties of the hypergeometric functions.The second order linear differential equationdyd2 yz(1 z) 2 [c (a b 1)z] aby 0,dzdzin which a, b and c are real or complex parameters, is called the hypergeometric equation. Series solutions of the hypergeometric equation valid in the neighborhoods ofz 0, 1 or can be developed by using Frobenious series method. Thus, if c is notan integer, the general solution of differential equation valid in a neighborhood of theorigin is found to bey A 2 F1 (a, b; c; z) Bz 1 c 2 F1 (a c 1, b c 1; 2 c; z),where A and B are arbitrary constants, and2 F1 (a, b; c; z) 1 aba(a 1)b(b 1) 2z z .c.1c(c 1).1.2 X(a)n (b)n z nn 0(c)nn!,(c 6 0, 1, 2, .)where (λ)ν denotes the Pochhammer symbol defined by(λ)0 1 and (λ)ν : Γ(λ ν).Γ(λ)Hence2 F1 (a, b; c; z) X(a)n (b)n z nn 08(c)nn!,

is called Gauss hypergeometric function. This series is convergent for z 1 whereRe(c) Re(b) 0 and z 1 where Re(c a b) 0.The Gauss hypergeometric function can be given by Euler’s integral representation asfollows:Γ (c)2 F1 (a, b; c; z) Γ(b)Γ(c b)1Ztb 1 (1 t)c b 1 (1 zt) a dt,0( z 1; Re(c) Re(b) 0).Replacing z zand by letting b , in Gauss’s hypergeometric equation, webhavezd2 ydy (c z) ay 0.dz 2dzThis equation has a regular singularity at z 0. The simplest solution of the equationisaa(a 1) 2z z .c.1c(c 1).1.2φ(a; c; z) 1 X(a)n z nn 0(c)n n!,(c 6 0, 1, 2, .).Hence, we getφ(a; c; z) X(a)n z nn 0(c)n n!,which is called confluent hypergeometric function.The confluent hypergeometric function can be given by an integral representation asfollows:Γ (c)φ(a; c; z) Γ(a)Γ(c a)Z1ta 1 (1 t)c a 1 exp (zt) dt,0(Re(c) Re(a) 0).9

A generalized form of the hypergeometric function is X(α1 )n .(αp )n z np Fq (α1 , .αp ; γ1 , ., γq ; z) n 0(γ1 )n .(γq )n n!,(2.5)(p, q 0, 1, .).Setting p 2, q 1 in (2.5), we get the Gauss hypergeometric function,F (α1 , α2 ; γ1 ; z) : 2 F1 (α1 , α2 ; γ1 ; z) X(α1 )n (α2 )n z n(γ1 )nn 0n!.Setting p q 1 in (2.5), we get confluent hypergeometric function,φ(α1 ; γ1 ; z) 1 F1 (α1 ; γ1 ; z) X(α1 )n z nn 0(γ1 )n n!.For the convergence of the series p Fq (see [28], [29]);case 1: If p q, then the series converges for all z;case 2: If p q 1, then the series converges for z 1 and otherwise series diverges;case 3: If p q 1, the series diverges for z 6 0.If the series terminates, there is no question of convergence, and the conclusions (case2) and (case 3) do not apply. If p q 1, then the series is absolutely convergent onthe circle z 1 ifReqXγj j 1pX!αi 0.i 1In 2004, Chaudhry et al. [5] used extended beta function Bp (x, y) to extend the hypergeometric functions (and confluent hypergeometric functions) as follows:Fp (a, b; c; z) XBp (b n, c b)n 0B (b, c b)(a)n(p 0 ; Re (c) Re (b) 0),10znn!

φp (b; c; z) XBp (b n, c b) z nn 0B (b, c b)n!(p 0 ; Re (c) Re (b) 0),and gave the Euler type integral representations1Fp (a, b; c; z) B (b, c b)1Zb 1t(1 t)c b 1 a(1 zt) exp 0 pdtt(1 t)(2.6)(p 0 ; p 0 and arg (1 z) π p; Re (c) Re (b) 0),andexp(z)φp (b; c; z) B (b, c b)1Zc b 1t(1 t)0b 1 pdtexp zt t(1 t) (2.7)(p 0 ; p 0 and Re (c) Re (b) 0).They called these functions as extended Gauss hypergeometric function (EGHF) andextended confluent hypergeometric function (ECHF), respectively, since F0 (a, b; c; z) 2 F1(a, b; c; z) and φ0 (b; c; z) 1 F1(b; c; z). They have discussed the differentia-tion properties and Mellin transforms of Fp (a, b; c; z) and φp (b; c; z) . They obtainedtransformation formulas, recurrence relations, summation and asymptotic formulas forthese functions.2.3Some Hypergeometric Functions of two and more variablesThere are four Appell’s hypergeometric functions of two variables (see [13], [31]). Inthis thesis we consider the first two Appell’s hypergeometric functions. These functions are defined by XB(a m n, d a)xn y mF1 (a, b, c; d; x, y) : (b)n (c)mB(a, d a)n! m!n,m 0(max { x , y } 1),11

and X(a)m n B (b n, d b) B (c m, e c) xn y mF2 (a, b, c; d, e; x, y) : B (b, d b) B (c; e c)n! m!n,m 0( x y 1).Lauricella functions of three variables are defined byFD3 XB (a m n r, e a) (b)m (c)n (d)r xm y n z r(a, b, c, d; e; x, y, z) : B (a, e a)m! n! r!m,n,r 0ppp( x y z 1).The Appell’s hypergeometric functions F1 and F2 can be given an integral representation as follows:F1 (a, b, c; d; x, y) Z 1Γ (d)Γ (a) Γ (d a)ta 1 (1 t)d a 1 (1 xt) b (1 yt) c dt,0( arg (1 x) π, arg (1 y) π; Re (d) Re (a) 0).and1F2 (a, b, c; d, e; x, y; p) B (b, d b) B (c, e c)Z1Z001 b 1t(1 t)d b 1 sc 1 (1 s)e c 1(1 xt ys)a( x y 1);(Re (d) Re (b) 0, Re (e) Re (c) 0, Re (a) 0).2.4Generating FunctionsIn this section, we give definitions of linear, bilinear, bilateral, multivariable, multilinear, multilateral and multiple generating functions, from the book [32].Definition 2.4.1. (Linear generating functions) If two-variable function F (x, t) canbe expanded as a formal power series expansion in t asF (x, t) Xn 012fn (x) tn

where each member of the coefficient set {fn (x)} n 0 is independent of t, then theF (x, t) is said to have generated the set {fn (x)}. Therefore F (x, t) is called a lineargenerating function for the set {fn (x)} .Definition 2.4.2. (Bilinear generating functions) If three-variable function F (x, y, t)can be expanded as a formal power series expansion in t such thatF (x, y, t) Xγn fn (x) fn (y) tnn 0where the sequence {γn } is independent of x,y and t, then F (x, y, t) is called a bilineargenerating function for the set {fn (x)} .Definition 2.4.3. (Bilateral generating functions) If three-variable function H (x, y, t)which is defined by a formal power series expansion in t asH (x, y, t) Xhn fn (x) gn (y) tnn 0where the sequence {hn } is independent of x, y and t, and the sets of functions {fn (x)} n 0and {gn (x)} n 0 are different, then H (x, y, t) is called a bilateral generating functionfor the set {fn (x)} or {gn (x)} .Definition 2.4.4. (Multivariable generating functions) Suppose that G (x1 , ., xr ; t) isa function of r 1 variables, which is defined by a formal power series expansion in tsuch thatG (x1 , ., xr ; t) Xcn gn (x1 , ., xr ) tnn 0where the sequence {cn } is independent of the variables x1 , ., xr and t. Then G (x1 , ., xr ; t)is called a multivariable generating function for the set {gn (x1 , ., xr )} n 0 corresponding to the nonzero coefficients cn .Definition 2.4.5. (Multilinear and multilateral generating functions) A multivariablegenerating functions G (x1 , ., xr ; t) which is defined in previous definition, is said tobe a multilinear generating function ifgn (x1 , ., xr ) fα1 (n) (x1 ).fαr (n) (xr )13

where the sequence α1 (n), ., αr (n) are functions of n which are not necessarily equal. If the functions fα1 (n) (x1 ) , . fαr (n) (xr ) are all different, the multivariable generating function G (x1 , ., xr ; t) is called a multilateral generating function.Definition 2.4.6. (Multiple generating functions) An extension of the multivariablegenerating function is said to be a multiple generating function which is defined formally byΨ (x1 , ., xr ; t1 , ., tr ) XC(n1 , .nr ) n1 ,.,nr (x1 , ., xr ) tn1 1 .tnr rn1 ,.,nr 0where the multiple sequence {C(n1 , .nr )} is independent of the variables x1 , ., xrand t1 , .tr .2.5Mellin Transform and Riemann-Liouville Fractional DerivativeIn this section, we give the definition of the Mellin transform and fractional derivativeoperator.Definition 2.5.1. Let f (x) be a function defined on the positive real axis 0 x .The Mellin transformation M is the operation, mapping the function f into the functionF , defined on the complex plane by the relation:Z M {f : s} F (s) xs 1 f (x)dx.0The function F (s) is called the Mellin transform of f.Example 2.5.2. The Mellin transform of the function f (x) exp( px) (p 0) isM {f : s} p s Γ(s)(Re(s) 0).Indeed, in order to obtain the Mellin transform of f (x) exp( px), we multiply bothsides of function f (x) by ps 1 and integrate with respect to p over the interval [0, ).14

ThusZ xs 1 exp( px)dx.M {f : s} F (s) 0Using the definition of the Gamma function, we getM {f : s} p s Γ(s).Definition 2.5.3. (Mellin inversion formula) The inverse transform of the Mellin transform is given byM 11{F : x} f (x) 2πiZi x s F (s)ds. i Definition 2.5.4. [16] The Riemann-Liouville fractional derivative of order µ is defined byDzµ {fZ1(z)} Γ ( µ)zf (t) (z t) µ 1 dt,0(Re (µ) 0)where the integration path is a line from 0 to z in complex t plane. For the casem 1 Re (µ) m (m 1, 2, 3.) , it is defined bydm µ mD{f (z)}dz m z Z zdm1 µ m 1f (t) (z t)dt . mdzΓ ( µ m) 0Dzµ {f (z)} Example 2.5.5. Let Re (λ) 1, Re (µ) 0. ThenDzµ {z λ } Z z1tλ (z t) µ 1 dtΓ ( µ) 0Z 11(zu)λ (z) µ 1 (1 u) µ 1 duΓ ( µ) 0Z 1z λ µuλ (1 u) µ 1 duΓ ( µ) 0B (λ 1, µ) λ µΓ (λ 1)z z λ µ .Γ ( µ)Γ (λ µ 1)Example 2.5.6. Let Re (λ) 0, Re (µ) 0 and z 1. ThenDzλ µ {z λ 1 (1 z) α } Γ (λ) µ 1z2 F1 (α, λ; µ; z) .Γ (µ)15

Solution. Direct calculations yieldDzλ µ {z λ 1 α(1 z)1} Γ (µ λ)zZtλ 1 (1 t) α (z t)µ λ 1 dt0 µ λ 1zzt αλ 1 1 t(1 t)dtΓ (µ λ) 0zZz µ λ 1 z λ 1 λ 1 u(1 uz) α (1 u)µ λ 1 duΓ (µ λ) 0Γ (λ) µ 1z 2 F1 (α, λ; µ; z) .Γ (µ)µ λ 1ZHence the proof is completed.2.6Elementary Series IdentityIn this section, we give general series identities which are used throughout the thesis.Lemma 2.6.1. [28] The following series identities X X XnXA(k, n) n 0 k 0n 0 k 0 XnX X XA(k, n k)(2.8)C(k, n k)(2.9)andC(k, n) n 0 k 0n 0 k 0holds.Proof. Consider the series X XA(k, n)tn k(2.10)n 0 k 0in which tn k has been inserted for convenience and will be removed later by takingt 1. Introducing new indices of summation j and m byn m j,k j,(2.11)so that the exponent (n k) in (2.10) becomes m. Since n 0 and k 0 in (2.11)then j 0, m j 0 or m j 0. Thus we arrive at X Xn 0 k 0n kA(k, n)t XmXm 0 j 016A(j, m j)tm .(2.12)

Finally, putting t 1 in the equality (2.12) and replacing j and m by k and n, we get(2.8). Similarly, equation (2.9) follows from (2.8).Lemma 2.6.2. For a bounded sequence {f (N )} N 0 of essentially arbitrary complexnumbers, we have X XX(x y)Nxn y kf (N ) .f (n k)N!n! k!n 0 k 0N 0(2.13)Proof. Using the Lemma 2.6.1, we get X X XNXxn y kxN k y kf (n k)(n k)! N!f (N )n! k!(N k)! k!n 0 k 0N 0 k 0 n XXN N k k f (N )xykN 0k 0 Xf (N ) (x y)N .N 0Whence the result.2.7Some Important PolynomialsIn this section, we consider some important polynomials which will be used in Chapter5.Definition 2.7.1. Lagrange polynomials in two variables are defined bynXxn k y k(α,β)(α)n k (β)k.gn (x, y) (n k)!k!k 0Definition 2.7.2. Lagrange-Hermite polynomials of two variables are defined by[ n2 ]Xxn 2k y kh(α,β)(x,y) .(α)(β)n 2kkn(n 2k)!k!k 0Definition 2.7.3. Hermite-Kampé de Feriét polynomials of two variables are definedby[ n2 ]Xxn 2k y kHn (x, y) .(n 2k)!k!k 0Definition 2.7.4. [35] Jacobi polynomials are defined bynX( 1)n k (β 1)n (α β 1)n k kn!(α,β)z .Pn (z) (α β n 1)n k 0 k! (n k)! (β 1)k (α β 1)n17

Chapter 3SOME GENERATING RELATIONS FOREXTENDED HYPERGEOMETRIC FUNCTIONS VIAGENERALIZED FRACTIONAL DERIVATIVEOPERATOR3.1IntroductionRecently an extension of beta function containing an extra parameter, which proved tobe useful earlier, was used to extend the hypergeometric functions [5]. The aim of thischapter is to obtain some generating functions for extended hypergeometric functions(EHF) by considering a new fractional derivative operator. We organize this chapter asfollows.In the second section, extensions of the first two Appell’s hypergeometric functions oftwo variables, namely, F1 (a, b, c; d; x, y; p) and F2 (a, b, c; d, e; x, y; p), and extended3(a, b, c, d; e; x, y, z) , areLauricella’s hypergeometric function of three variables, FD,pdefined and integral representations of F1 (a, b, c; d; x, y; p) and F2 (a, b, c; d, e; x, y; p)are obtained. In the third section, extended fractional derivative operator is definedand extended fractional derivatives of some elementary functions, which are neededin obtaining generating functions, are calculated in terms of extended Appell’s hypergeometric functions and extended Lauricella’s hypergeometric function. In the fourthsection, some results related with Mellin transforms and extended fractional derivative operators are given. Last section contains the main results of the chapter. In thissection, linear and bilinear generating functions for the extended hypergeometric functions are obtained via generalized fractional derivative operator by following the same18

method explained in [32].3.2The E

TABLE OF CONTENTS ABSTRACT . Mff: sg Mellin transform of f viii. Chapter 1 INTRODUCTION Many important functions in applied sciences are defined via improper integrals or se-ries (or infinite products). The general name of these important functions are called special fun