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Thabet Abdeljawad - One of the best experts on this subject based on the ideXlab platform.
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properties and applications of a new extended gamma function involving confluent hypergeometric function
Journal of Mathematics, 2021Co-Authors: Abdus Saboor, Kottakkaran Sooppy Nisar, Gauhar Rahman, Hazrat Ali, Thabet AbdeljawadAbstract:In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer Symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.
Jyotindra C Prajapati - One of the best experts on this subject based on the ideXlab platform.
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on a generalization of mittag leffler function and its properties
Journal of Mathematical Analysis and Applications, 2007Co-Authors: Ajay K Shukla, Jyotindra C PrajapatiAbstract:Abstract Let s and z be complex variables, Γ ( s ) the Gamma function, and ( s ) ν = Γ ( s + ν ) Γ ( s ) for any complex ν the generalized Pochhammer Symbol. The principal aim of the paper is to investigate the function E α , β γ , q ( z ) = ∑ n = 0 ∞ ( γ ) q n Γ ( α n + β ) z n n ! , where α , β , γ ∈ C ; Re ( α ) > 0 , Re ( β ) > 0 , Re ( γ ) > 0 and q ∈ ( 0 , 1 ) ∪ N . This is a generalization of the exponential function exp ( z ) , the confluent hypergeometric function Φ ( γ , α ; z ) , the Mittag-Leffler function E α ( z ) , the Wiman's function E α , β ( z ) and the function E α , β γ ( z ) defined by Prabhakar. For the function E α , β γ , q ( z ) its various properties including usual differentiation and integration, Laplace transforms, Euler (Beta) transforms, Mellin transforms, Whittaker transforms, generalised hypergeometric series form, Mellin–Barnes integral representation with their several special cases are obtained and its relationship with Laguerre polynomials, Fox H-function and Wright hypergeometric function is also established.
Kottakkaran Sooppy Nisar - One of the best experts on this subject based on the ideXlab platform.
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properties and applications of a new extended gamma function involving confluent hypergeometric function
Journal of Mathematics, 2021Co-Authors: Abdus Saboor, Kottakkaran Sooppy Nisar, Gauhar Rahman, Hazrat Ali, Thabet AbdeljawadAbstract:In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer Symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.
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a new extension of the Pochhammer Symbol and its application to hypergeometric functions
International Journal of Applied and Computational Mathematics, 2019Co-Authors: Maryam Safdar, Gauhar Rahman, Zafar Ullah, Abdul Ghaffar, Kottakkaran Sooppy NisarAbstract:Our main goal in this present paper is to define first a new extension of the Pochhammer Symbol and the gamma functions which involving the Mittag-Leffler function in their kernels. By using this extended Pochhammer Symbol, we then introduce and investigate the corresponding extension of the generalized hypergeometric functions and of some of its special cases. Also, we establish the basic properties and results for the extended $$\tau $$-Gauss hypergeometric function, which includes integral and derivative formulas involving the Mellin transform and fractional calculus approaches. Some new and known results as consequences of our proposed extension of the $$\tau $$-Gauss hypergeometric function are also established.
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some extensions of the Pochhammer Symbol and the associated hypergeometric functions
Iranian Journal of Science and Technology Transaction A-science, 2019Co-Authors: H M Srivastava, Gauhar Rahman, Kottakkaran Sooppy NisarAbstract:In the present paper, we first define an extended Pochhammer Symbol by using a known extension of the gamma function involving the modified Bessel (or Macdonald) function. By using this extended Pochhammer Symbol, we then introduce and investigate the corresponding extension of the generalized hypergeometric function and of some of its special cases. We also present some families of generating functions and generating relations for the extended hypergeometric functions.
H M Srivastava - One of the best experts on this subject based on the ideXlab platform.
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some extensions of the Pochhammer Symbol and the associated hypergeometric functions
Iranian Journal of Science and Technology Transaction A-science, 2019Co-Authors: H M Srivastava, Gauhar Rahman, Kottakkaran Sooppy NisarAbstract:In the present paper, we first define an extended Pochhammer Symbol by using a known extension of the gamma function involving the modified Bessel (or Macdonald) function. By using this extended Pochhammer Symbol, we then introduce and investigate the corresponding extension of the generalized hypergeometric function and of some of its special cases. We also present some families of generating functions and generating relations for the extended hypergeometric functions.
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a certain generalized Pochhammer Symbol and its applications to hypergeometric functions
Applied Mathematics and Computation, 2014Co-Authors: H M Srivastava, Aysegul Cetinkaya, Onur I KiymazAbstract:In this article, we first introduce an interesting new generalization of the familiar Pochhammer Symbol by means of a certain one-parameter family of generalized gamma functions. With the help of this new generalized Pochhammer Symbol, we then introduce an extension of the generalized hypergeometric function "rF"s with r numerator and s denominator parameters. Finally, we present a systematic study of the various fundamental properties of the class of the generalized hypergeometric functions introduced here.
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generating relations and other results associated with some families of the extended hurwitz lerch zeta functions
SpringerPlus, 2013Co-Authors: H M SrivastavaAbstract:Throughout our present investigation, we use the following standard notations: ℕ: = {1, 2, 3, ⋯ }, ℕ0: = {0, 1, 2, 3, ⋯ } = ℕ ∪ {0} and Z−:={−1,−2,−3,⋯}=Z0−∖{0}. Here, as usual,ℤ denotes the set of integers,ℝ denotes the set of real numbers, ℝ+ denotes the set of positive real numbers andℂ denotes the set of complex numbers. The familiar general Hurwitz-Lerch Zeta function Φ(z, s, a) defined by (see, for example, (Erdelyi et al.1953, p. 27. Eq. 1.11 (1)); see also Srivastava and Choi ((2001, p. 121 et seq.) and (Srivastava and Choi 2012), p. 194 et seq.) Φ(z,s,a):=∑n=0∞zn(n+a)sa∈C∖Z0−;s∈Cwhen|z| 1when|z|=1 1.1 contains, as its special cases, not only the Riemann Zeta function ζ(s), the Hurwitz (or generalized) Zeta function ζ(s, a) and the Lerch Zeta function ls(ξ) defined by (see, for details, (Erdelyi et al.1953, Chapter I) and Srivastava and Choi ((2001), Chapter 2) ζ(s):=∑n=0∞1(n+1)s=Φ(1,s,1)=ζ(s,1)ℜ(s)>1, 1.2 ζ(s,a):=∑n=0∞1(n+a)s=Φ(1,s,a)ℜ(s)>1;a∈C∖Z0− 1.3 and ls(ξ):=∑n=0∞e2nπiξ(n+1)s=Φe2πiξ,s,1ℜ(s)>1;ξ∈R, 1.4 respectively, but also such other important functions of Analytic Number Theory as the Polylogarithmic function (or de Jonquiere’s function) Lis(z): Lis(z):=∑n=1∞znns=zΦ(z,s,1)s∈Cwhenz 1whenz=1 1.5 and the Lipschitz-Lerch Zeta function ϕ(ξ, a, s) (see Srivastava and Choi ((2001), p. 122, Equation 2.5 (11))): ϕ(ξ,s,a):=∑n=0∞e2nπiξ(n+a)s=Φe2πiξ,s,aa∈C∖Z0−;ℜ(s)>0whenξ∈R∖Z;ℜ(s)>1whenξ∈Z, 1.6 which was first studied by Rudolf Lipschitz (1832-1903) and Matyas Lerch (1860-1922) in connection with Dirichlet’s famous theorem on primes in arithmetic progressions (see also (Srivastava 2011), Section 5). Indeed, just as its aforementioned special cases ζ(s) and ζ(s, a), the Hurwitz-Lerch Zeta function Φ(z, s, a) defined by (1.1) can be continued meromorphically to the whole complex s-plane, except for a simple pole at s = 1 with its residue 1. It is also known that (Erdelyi et al.1953, p. 27, Equation 1.11 (3)) Φ(z,s,a)=1Γ(s)∫0∞ts−1e−at1−ze−tdt=1Γ(s)∫0∞ts−1e−(a−1)tet−zdtℜ(a)>0;ℜ(s)>0when|z|≦1(z≠1);ℜ(s)>1whenz=1. 1.7 Making use of the Pochhammer Symbol (or the shifted factorial) (λ)ν (λ, ν ∈ ℂ) defined, in terms of the familiar Gamma function, by (λ)ν:=Γ(λ+ν)Γ(λ)=1(ν=0;λ∈C∖{0})λ(λ+1)⋯(λ+n−1)(ν=n∈N;λ∈C), 1.8 it being understood conventionally that (0)0: = 1 and assumed tacitly that the Gamma quotient exists, we recall each of the following well-known expansion formulas: ζ(s,a−t)=∑n=0∞(s)nn!ζ(s+n,a)tn(|t|<|a|) 1.9 and Φ(z,s,a−t)=∑n=0∞(s)nn!Φ(z,s+n,a)tn(|t|<|a|). 1.10 More generally, it is not difficult to show similarly that ∑n=0∞(λ)nn!Φ(z,s+n,a)tn=∑k=0∞zk(k+a)s−λ(k+a−t)λ=:ϑλ(z,t;s,a)(|t|<|a|), 1.11 which would reduce immediately to the expansion formula (1.10) in its special case when λ = s. Moreover, in the limit case when t↦tλand|λ|→∞, this last result (1.11) yields ∑n=0∞Φ(z,s+n,a)tnn!=∑k=0∞zk(k+a)sexptk+a=:φ(z,t;s,a)(|t|<∞). 1.12 Wilton (1922/1923) applied the expansion formula (1.9) in order to rederive Burnside’s formula (Erdelyi et al.1953, p. 48, Equation 1.18 (11)) for the sum of a series involving the Hurwitz (or generalized) Zeta function ζ(s, a). Srivastava (see, for details, Srivastava (1988a;1988b)), on the other hand, made use of such expansion formulas as (1.9) and (1.10) as well as the obvious special case of (1.9) when a=1 for finding the sums of various classes of series involving the Riemann Zeta function ζ(s) and the Hurwitz (or generalized) Zeta function ζ(s, a) (see also Srivastava and Choi ((2001), Chapter 3) and (Srivastava and Choi 2012), Chapter 3). Various results for the generating functions ϑλ(z, t;s, a) and φ(z, t;s, a), which are defined by (1.11) and (1.12), respectively, were given recently by Bin-Saad (2007, p. 46, Equations (5.1) to (5.4)) who also considered each of the following truncated forms of these generating functions: ϑλ(0,r)(z,t;s,a):=∑k=0rzk(k+a)s−λ(k+a−t)λ(r∈N0), 1.13 ϑλ(r+1,∞)(z,t;s,a):=∑k=r+1∞zk(k+a)s−λ(k+a−t)λ(r∈N0), 1.14 φ(0,r)(z,t;s,a):=∑k=0rzk(k+a)sexptk+a(r∈N0) 1.15 and φ(r+1,∞)(z,t;s,a):=∑k=r+1∞zk(k+a)sexptk+a(r∈N0), 1.16 so that, obviously, we have ϑλ(0,r)(z,t;s,a)+ϑλ(r+1,∞)(z,t;s,a)=ϑλ(z,t;s,a) 1.17 and φ(0,r)(z, t; s, a) + φ(r+1,∞)(z, t; s, a) = φ(z, t; s, a). 1.18 For the Riemann Zeta function ζ(s), the special case of each of the generating functions ϑλ(z, t;s, a) and φ(z, t;s, a) in (1.11) and (1.12) when z = a = 1 was investigated by Katsurada (1997). Subsequently, various results involving the generating functions ϑλ(z, t;s, a) and φ(z, t;s, a) defined by (1.11) and (1.12), respectively, together with their such partial sums as those given by (1.13) to (1.16), were derived by Bin-Saad (2007) (see also the more recent sequels to (Bin-Saad 2007) and (Katsurada 1997) by Gupta and Kumari (2011) and by Saxena et al. (2011a). Our main objective in this paper is to investigate, in a rather systematic manner, much more general families of generating functions and their partial sums than those associated with the generating functions ϑλ(z, t;s, a) and φ(z, t;s, a) defined by (1.11) and (1.12), respectively. We also show the hitherto unnoticed fact that the so-called τ-generalized Riemann Zeta function, which happens to be the main subject of investigation by Gupta and Kumari (2011) and by Saxena et al. (2011a), is simply a seemingly trivial notational variation of the familiar general Hurwitz-Lerch Zeta function Φ(z, s, a) defined by (1.1). Finally, we present a sum-integral representation formula for the general family of the extended Hurwitz-Lerch Zeta functions.
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fractional calculus with an integral operator containing a generalized mittag leffler function in the kernel
Applied Mathematics and Computation, 2009Co-Authors: H M Srivastava, Ivorad TomovskiAbstract:In this paper, we introduce and investigate a fractional calculus with an integral operator which contains the following family of generalized Mittag-Leffler functions:E"@a","@b^@c^,^@k(z)[email protected]?n=0~(@c)"@k"[email protected](@[email protected])z^nn!(z,@b,@[email protected]?C;R(@a)>max{0,R(@k)-1};R(@k)>0)in its kernel, (@l)"@n being the familiar Pochhammer Symbol. A number of corollaries and consequences of the main results presented here are also considered.
Gauhar Rahman - One of the best experts on this subject based on the ideXlab platform.
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properties and applications of a new extended gamma function involving confluent hypergeometric function
Journal of Mathematics, 2021Co-Authors: Abdus Saboor, Kottakkaran Sooppy Nisar, Gauhar Rahman, Hazrat Ali, Thabet AbdeljawadAbstract:In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer Symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.
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a new extension of the Pochhammer Symbol and its application to hypergeometric functions
International Journal of Applied and Computational Mathematics, 2019Co-Authors: Maryam Safdar, Gauhar Rahman, Zafar Ullah, Abdul Ghaffar, Kottakkaran Sooppy NisarAbstract:Our main goal in this present paper is to define first a new extension of the Pochhammer Symbol and the gamma functions which involving the Mittag-Leffler function in their kernels. By using this extended Pochhammer Symbol, we then introduce and investigate the corresponding extension of the generalized hypergeometric functions and of some of its special cases. Also, we establish the basic properties and results for the extended $$\tau $$-Gauss hypergeometric function, which includes integral and derivative formulas involving the Mellin transform and fractional calculus approaches. Some new and known results as consequences of our proposed extension of the $$\tau $$-Gauss hypergeometric function are also established.
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some extensions of the Pochhammer Symbol and the associated hypergeometric functions
Iranian Journal of Science and Technology Transaction A-science, 2019Co-Authors: H M Srivastava, Gauhar Rahman, Kottakkaran Sooppy NisarAbstract:In the present paper, we first define an extended Pochhammer Symbol by using a known extension of the gamma function involving the modified Bessel (or Macdonald) function. By using this extended Pochhammer Symbol, we then introduce and investigate the corresponding extension of the generalized hypergeometric function and of some of its special cases. We also present some families of generating functions and generating relations for the extended hypergeometric functions.
