$q,\omega $-Tricomi expansions and exponential integral associated with Hahn difference operator

Abstract

Abstract In this paper, we introduce a $q,\omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:math> -analog of Tricomi expansion based on Hahn’s difference operator. Some properties of $q,\omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:math> -Tricomi expansion are derived and proved in terms of incomplete $q,\omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:math> -gamma functions. Also, a $q,\omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:math> -analog of the exponential integral is presented as a series expansion of incomplete $q,\omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:math> -gamma functions and shown to be a limiting case of a $q,\omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:math> -Tricomi expansion.