Some Symmetric Identities for the Multiple (<i>p</i>, <i>q</i>)-Hurwitz-Euler eta Function

• Main Authors: Kyung-Won Hwang, Cheon Seoung Ryoo
• Format: Article
• Language: English
• Published: MDPI AG 2019-05-01
• Series: Symmetry
• Subjects: Euler numbers and polynomials; <i>q</i>-Euler numbers and polynomials; Hurwitz-Euler eta function; multiple Hurwitz-Euler eta function; higher order <i>q</i>-Euler numbers and polynomials; (<i>p</i>, <i>q</i>)-Euler numbers and polynomials of higher order; symmetric identities; symmetry of the zero
• Online Access: https://www.mdpi.com/2073-8994/11/5/645
• ISSN: 2073-8994

The main purpose of this paper is to find some interesting symmetric identities for the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Hurwitz-Euler eta function by generalizing the Carlitz&#8217;s form <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Euler numbers and polynomials. We find some formulas and properties involved in Carlitz&#8217;s form <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Euler numbers and polynomials with higher order. We find new symmetric identities for multiple <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz&#8217;s form <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Euler numbers and polynomials with higher order by using symmetry about multiple <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz&#8217;s form <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-Euler numbers and polynomials with higher order.