scholarly journals Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 483 ◽  
Author(s):  
Mehmet Ali Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Beta Function ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integral Operators ◽  
Derivative Formulas ◽  
Generating Relations ◽  
Transformation Formulas

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( x , z ) . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

scholarly journals Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan ◽  
Salah Mahmoud Boulaaras ◽  
Bahri-Belkacem Cherif
Keyword(s):  
Probability Theory ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integrals ◽  
Convergence Properties ◽  
Fractional Integral Operators ◽  
Derivative Formulas

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.

scholarly journals A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
KS Nisar ◽  
Shahid Mubeen
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Integral Representations ◽  
Basic Properties ◽  
Mellin Transformation ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

Fractional integral operators characterized by some new hypergeometric summation formulas

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Min-Jie Luo ◽  
Ravinder Krishna Raina
Keyword(s):  
Special Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Fractional Integral Operators ◽  
Summation Formulas ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

AbstractThe purpose of this paper is to study generalized fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions

scholarly journals Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Eze R. Nwaeze ◽  
Muhammad Adil Khan ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Integral ◽  
Convex Set ◽  
Beta Function ◽  
Integral Operators ◽  
Fractional Integral Operators ◽  
Interval Valued Function ◽  
Class Of Functions ◽  
Interval Valued

Abstract The notion of m-polynomial convex interval-valued function $\Psi =[\psi ^{-}, \psi ^{+}]$ Ψ = [ ψ − , ψ + ] is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions $\psi ^{-}$ ψ − and $\psi ^{+}$ ψ + . For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, $\rho,\epsilon >0$ ρ , ϵ > 0 and $\zeta,\eta \in {\mathbf{S}}$ ζ , η ∈ S , then $$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$ m m + 2 − m − 1 Ψ ( ζ + η 2 ) ⊇ Γ ρ ( ϵ + ρ ) ( η − ζ ) ϵ ρ [ ρ J ζ + ϵ Ψ ( η ) + ρ J η − ϵ Ψ ( ζ ) ] ⊇ Ψ ( ζ ) + Ψ ( η ) m ∑ p = 1 m S p ( ϵ ; ρ ) , where Ψ is Lebesgue integrable on $[\zeta,\eta ]$ [ ζ , η ] , $S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )$ S p ( ϵ ; ρ ) = 2 − ϵ ϵ + ρ p − ϵ ρ B ( ϵ ρ , p + 1 ) and $\mathcal{B}$ B is the beta function. We extend, generalize, and complement existing results in the literature. By taking $m\geq 2$ m ≥ 2 , we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.

scholarly journals A NEW EXTENSION OF THE HURWITZ- LERCH ZETA FUNCTION AND PROPERTIES USING THE EXTENDED BETA FUNCTION \(B_{p,q}^{(ρ,σ,τ)}(x,y)\)

2020 ◽  
Vol 1 (3) ◽  
pp. 120-127
Author(s):  
Salem Saleh Barahmah
Keyword(s):  
Zeta Function ◽  
Recurrence Relations ◽  
Beta Function ◽  
Integral Representations ◽  
Lerch Zeta Function ◽  
Generating Relations ◽  
Extended Beta Function

The purpose of present paper is to introduce a new extension of Hurwitz-Lerch Zeta function by using the extended Beta function. Some recurrence relations, generating relations and integral representations are derived for that new extension.

scholarly journals A new extension of Srivastava’s triple hypergeometric functions and their associated properties

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdus Saboor ◽  
Gauhar Rahman ◽  
Zunaira Anjum ◽  
Kottakkaran Sooppy Nisar ◽  
Serkan Araci
Keyword(s):  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Integral Representations ◽  
Pochhammer Symbol ◽  
Basic Properties ◽  
Special Cases ◽  
Derivative Formulas

AbstractIn this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

scholarly journals A Note on the New Extended Beta Function with Its Application

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Vandana Palsaniya ◽  
Ekta Mittal ◽  
Sunil Joshi ◽  
D. L. Suthar
Keyword(s):  
Hypergeometric Function ◽  
Beta Function ◽  
Moment Generating Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Summation Formulas ◽  
Derivative Formulas ◽  
New Type ◽  
Extended Beta Function ◽  
Transformation Formulas

The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function and ℓ .

scholarly journals Extended incomplete version of hypergeometric functions

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 653-662
Author(s):  
Mehmet Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integral Representations ◽  
Elementary Functions ◽  
Gamma Functions ◽  
Mellin Transforms ◽  
Incomplete Gamma Functions ◽  
Derivative Formulas ◽  
Transformation Formulas

Recently, the incomplete Pochhammer ratios are defined in terms of incomplete beta and gamma functions [10]. In this paper, we introduce the extended incomplete version of Pochhammer symbols in terms of the generalized incomplete gamma functions. With the help of this extended incomplete version of Pochhammer symbols we introduce the extended incomplete version of Gauss hypergeometric and Appell?s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, Mellin transforms and log convex properties. Furthermore, we investigate incomplete fractional derivatives for extended incomplete version of some elementary functions.

scholarly journals Some Results of Extended Beta Function and Hypergeometric Functions by Using Wiman’s Function

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2944
Author(s):  
Shilpi Jain ◽  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Antonella Lupica ◽  
Clemente Cesarano
Keyword(s):  
Hypergeometric Function ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Gauss Hypergeometric Function ◽  
Logarithmic Convexity ◽  
Functional Relations ◽  
Derivative Formulas ◽  
Extended Beta Function ◽  
Transformation Formulas

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

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