scholarly journals The incomplete Srivastava’s triple hypergeometric functions γHB and ΓHB

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1779-1787 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh Parmar ◽  
Purnima Chopra
Keyword(s):  
Recurrence Relation ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Reduction Formula ◽  
Gamma Functions ◽  
Fundamental Properties ◽  
Special Cases ◽  
Incomplete Gamma Functions ◽  
Derivative Formula

Recently Srivastava et al. [26] introduced the incomplete Pochhammer symbols by means of the incomplete gamma functions ?(s,x) and ?(s,x), and defined incomplete hypergeometric functions whose a number of interesting and fundamental properties and characteristics have been investigated. Further, ?etinkaya [6] introduced the incomplete second Appell hypergeometric functions and studied many interesting and fundamental properties and characteristics. In this paper, motivated by the abovementioned works, we introduce two incomplete Srivastava?s triple hypergeometric functions ?HB and ?HB by using the incomplete Pochhammer symbols and investigate certain properties, for example, their various integral representations, derivative formula, reduction formula and recurrence relation. Various (known or new) special cases and consequences of the results presented here are also considered.

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 On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

scholarly journals On the integral representations for the confluent hypergeometric function

2016 ◽  
Vol 472 (2193) ◽  
pp. 20160421 ◽  
Author(s):  
Bujar Xh. Fejzullahu
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Contour Integral ◽  
Confluent Hypergeometric Functions ◽  
Special Cases

In this paper, we derive a new contour integral representation for the confluent hypergeometric function as well as for its various special cases. Consequently, we derive expansions of the confluent hypergeometric function in terms of functions of the same kind. Furthermore, we obtain a new identity involving integrals and sums of confluent hypergeometric functions. Our results generalized several well-known results in the literature.

scholarly journals Some applications of Srivastava’s theorem involving a certain family of generalized and extended hypergeometric polynomials

Filomat ◽  
2015 ◽  
Vol 29 (8) ◽  
pp. 1811-1819 ◽  
Author(s):  
Shy-Der Lin ◽  
H.M. Srivastava ◽  
Mu-Ming Wong
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Functions ◽  
Fundamental Properties ◽  
Hypergeometric Polynomials ◽  
Special Cases

Recently, Srivastava et al. [H. M. Srivastava, M. A. Chaudhry and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced and initiated the study of many interesting fundamental properties and characteristics of a certain pair of potentially useful families of the so-called generalized incomplete hypergeometric functions. Ever since then there have appeared many closely-related works dealing essentially with notable developments involving various classes of generalized hypergeometric functions and generalized hypergeometric polynomials, which are defined by means of the corresponding incomplete and other novel extensions of the familiar Pochhammer symbol. Here, in this sequel to some of these earlier works, we derive several general families of hypergeometric generating functions by applying Srivastava?s Theorem. We also indicate various (known or new) special cases and consequences of the results presented in this paper.

scholarly journals Some Characterizations of the Extended Beta and Gamma Functions: Properties and Applications

2021 ◽  
pp. 101-112
Keyword(s):  
Recurrence Relations ◽  
Integral Representations ◽  
General Introduction ◽  
Differentiation Formula ◽  
Gamma Functions ◽  
Definite Integrals ◽  
Special Cases ◽  
Form Representation ◽  
Basic Functions ◽  
Extended Function

The article represents the elementary and general introduction of some characterizations of the extended gamma and beta Functions and their important properties with various representations. This paper provides reviews of some of the new proposals to extend the form of basic functions and some closed-form representation of more integral functions is described. Some of the relative behaviors of the extended function, the special cases resulting from them when fixing the parameters, the decomposition equation, the integrative representation of the proposed general formula, the correlations related to the proposed formula, the frequency relationships, and the differentiation equation for these basic functions were investigated. We also investigated the asymptotic behavior of some special cases, known formulas, the basic decomposition equation, integral representations, convolutions, recurrence relations, and differentiation formula for these target functions by studying. Applications of these functions have been presented in the evaluation of some reversible Laplace transforms to the complex of definite integrals and the infinite series of related basic functions.

scholarly journals ON SUMS OF INDEPENDENT GENERALIZED PARETO RANDOM VARIABLES WITH APPLICATIONS TO INSURANCE AND CAT BONDS

2017 ◽  
Vol 32 (2) ◽  
pp. 296-305 ◽  
Author(s):  
Saralees Nadarajah ◽  
Yuanyuan Zhang ◽  
Tibor K. Pogány
Keyword(s):  
Random Variables ◽  
Exact Distribution ◽  
Integral Representations ◽  
Gamma Functions ◽  
Catastrophe Bonds ◽  
Generalized Pareto ◽  
Cat Bonds ◽  
Incomplete Gamma Functions ◽  
Single Integral

We derive single integral representations for the exact distribution of the sum of independent generalized Pareto random variables. The integrands involve the incomplete and complementary incomplete gamma functions. Applications to insurance and catastrophe bonds are described.

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 MORE SPECIAL FUNCTIONS TRAPPED

2013 ◽  
Vol 24 (02) ◽  
pp. 1350004
Author(s):  
CHARLES SCHWARTZ
Keyword(s):  
Hypergeometric Function ◽  
Bessel Functions ◽  
Special Functions ◽  
Mathematical Physics ◽  
Confluent Hypergeometric Function ◽  
Trapezoidal Rule ◽  
Integral Representations ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

We extend the technique of using the trapezoidal rule for efficient evaluation of the special functions of mathematical physics given by integral representations. This technique was recently used for Bessel functions, and here we treat incomplete gamma functions and the general confluent hypergeometric function.

scholarly journals Generating Functions for Some Hypergeometric Functions of Four Variables via Laplace Integral Representations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Jihad Younis ◽  
Maged Bin-Saad ◽  
Ashish Verma
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Essential Role ◽  
Integral Representations ◽  
Laplace Integral ◽  
Special Cases ◽  
Generating Relations

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered.

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