Inverse hyperbolic equation, Spectral technique, Regularization method, Operational matrix

Abstract

Recently, the realm related to Euler's Beta function has played a significant role in the development of special function theory. In this study, a new extension of the special function known as Euler's Beta function with respect to the Mittag-Leffler-Kummer function is introduced. Another formula of this new Beta function in terms of the Fox-Wright function is also presented. Numerous analytical properties of this new special function, such as several functional and summation relations, Mellin transforms, integral representations, and several derivative formulas, are studied. Furthermore, significant statistical implementations of Euler's Beta distribution are also discussed.

Keywords

• euler's integral function
• euler's beta function
• mittag-leffler function
• confluent hypergeometric function
• mellin transform
• 35q31
• 33e12
• 33c15
• 33b20
• 44a20