Some properties concerning the analysis of generalized Wright function

Abstract Solving a linear partial differential equation witness a noteworthy role of Wright function. Due to its usefulness and various applications, a variety of its extensions (and generalizations) have been investigated and presented. The purpose and design of the paper are intended to study and come up with a new extension of the generalized Wright function by using generalized beta function and obtain some integral representation of the freshly defined function. Also, we present the Mellin transform of this function in the form of Fox–Wright function. Furthermore, we obtain the recurrence relation, derivative formula for the said function and also by using an extended Riemann–Liouville fractional derivative, we present a fractional derivative formula for the generalized Wright function.

[1]  I. Podlubny.  Fractional differential equations , 1998 .

[2]  H. Flegg.  Mikusinski's Operational Calculus , 1974 .

[3]  F. Mainardi,et al.  The role of the Fox-Wright functions in fractional sub-diffusion of distributed order , 2007, 0711.3779.

[4]  Hari M. Srivastava,et al.  Extended hypergeometric and confluent hypergeometric functions , 2004, Appl. Math. Comput..

[5]  Ji-Hun Yoon.  Mellin Transform Method for European Option Pricing with Hull-White Stochastic Interest Rate , 2014, J. Appl. Math..

[6]  Francesco Mainardi.  On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$ , 2014 .

[7]  E. Wright.  The Asymptotic Expansion of the Generalized Hypergeometric Function , 1935 .

[8]  R. Gorenflo,et al.  Analytical properties and applications of the Wright function , 2007, math-ph/0701069.

[9]  M. El-shahed,et al.  An Extension of Wright Function and Its Properties , 2015 .

[10]  Jeong-Hoon Kim,et al.  The pricing of vulnerable options with double Mellin transforms , 2015 .

[11]  Myungjoo Kang,et al.  Valuing vulnerable geometric Asian options , 2016, Comput. Math. Appl..

[12]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[13]  M. Ali Özarslan,et al.  Extension of gamma, beta and hypergeometric functions , 2011, J. Comput. Appl. Math..

[14]  E. M. Wright,et al.  The asymptotic expansion of integral functions defined by Taylor series , 1940, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[15]  E. Wright.  On the Coefficients of Power Series Having Exponential Singularities , 1933 .

[16]  H. Srivastava,et al.  Zeta and q-Zeta Functions and Associated Series and Integrals , 2011 .

[17]  E. Wright.  THE GENERALIZED BESSEL FUNCTION OF ORDER GREATER THAN ONE , 1940 .

[18]  Myungjoo Kang,et al.  Pricing vulnerable path-dependent options using integral transforms , 2017, J. Comput. Appl. Math..