| Summary: | In this paper, we introduce the <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-analog of the <i>p</i>-adic factorial function. By utilizing some properties of <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-numbers, we obtain several new and interesting identities and formulas. We then construct the <i>p</i>-adic <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. We also derive more representations of the <i>p</i>-adic <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-gamma function in general case. Moreover, we consider the <i>p</i>-adic <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-Euler constant derived from the derivation of <i>p</i>-adic <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-gamma function at <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. Furthermore, we provide a limit representation of aforementioned Euler constant based on <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-numbers. Finally, we consider <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-extension of the <i>p</i>-adic beta function via the <i>p</i>-adic <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mi>ρ</mi> <mo>,</mo> <mi>q</mi> </mfenced> </semantics> </math> </inline-formula>-gamma function and we then investigate various formulas and identities.
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