Abstract:
Abstract This article explores some properties of degenerate hypergeometric Bernoulli polynomials, which are defined through the following generating function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msup> <m:msubsup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mi>λ</m:mi> <m:mi>x</m:mi> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msubsup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mi>λ</m:mi> <m:mi>x</m:mi> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>-</m:mo> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mi>l</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mi>l</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mi>l</m:mi> </m:msup> </m:mrow> <m:mrow> <m:mi>l</m:mi> <m:mo>!</m:mo> </m:mrow> </m:mfrac> </m:mrow> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:msup> <m:mrow/> <m:mo>∞</m:mo> </m:msup> </m:mrow> </m:munderover> <m:mrow> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:msubsup> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>!</m:mo> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi> </m:mi> <m:mi> </m:mi> <m:mrow> <m:mo>|</m:mo> <m:mi>t</m:mi> <m:mo>|</m:mo> </m:mrow> <m:mo><</m:mo> <m:mo>min</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mi>π</m:mi> <m:mo>,</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>λ</m:mi> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mfrac> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> <m:mo>\</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mn>0</m:mn> <m:mo>}</m:mo> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> {{{t^m}e_\lambda ^x\left( t \right)} \over {e_\lambda ^x\left( t \right) - \sum\nolimits_{l = 0}^{m - 1} {\left( 1 \right)l,\lambda{{{t^l}} \over {l!}}} }} = \sum\limits_{n = 0}^{^\infty } {B_{n,\lambda }^{\left[ {m - 1} \right]}} \left( x \right){{{t^n}} \over {n!}},\,\,\,\,\left| t \right| < \min \left\{ {2\pi ,{1 \over {\left| \lambda \right|}}} \right\},\lambda \in \mathbb{R}\backslash \left\{ 0 \right\}. We deduce their associated summation formulas and their corresponding determinant form. Also we focus our attention on the zero distribution of such polynomials and perform some numerical illustrative examples, which allow us to compare the behavior of the zeros of degenerate hypergeometric Bernoulli polynomials with the zeros of their hypergeometric counterpart. Finally, using a monomiality principle approach we present a differential equation satisfied by these polynomials.
Tópico:
Mathematical functions and polynomials
