Poisson like matrix operator and its application in <i>p</i>-summable space
Taja Yaying, Bipan Hazarika, Merve İlkhan, M. Mursaleen
Abstract
Abstract The incomplete gamma function Γ( a , u ) is defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mi mathvariant="normal">Γ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:munderover> <m:mo>∫</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>u</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munderover> <m:msup> <m:mi>t</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>a</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mtext>e</m:mtext> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo>−</m:mo> <m:mi>t</m:mi> </m:mrow> </m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mtext>d</m:mtext> </m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> </m:math> $$\Gamma(a,u)=\int\limits_{u}^{\infty}t^{a-1}\textrm{e}^{-t}\textrm{d} t,$$ where u > 0. Using the incomplete gamma function, we define a new Poisson like regular matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="fraktur">P</m:mi> </m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>p</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> <m:mi>k</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>μ</m:mi> </m:mrow> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:math> $\mathfrak{P}(\mu)=(p^{\mu}_{nk})$ given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msubsup> <m:mi>p</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> <m:mi>k</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>μ</m:mi> </m:mrow> </m:msubsup> <m:mo>=</m:mo> <m:mfenced open="{" close=""> <m:mtable columnalign="left left" rowspacing="0.53em 0.1em" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>!</m:mo> </m:mrow> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfrac> </m:mstyle> <m:mstyle> <m:mfrac> <m:mrow> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mtext>e</m:mtext> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo>−</m:mo> <m:mi>μ</m:mi> </m:mrow> </m:msup> <m:msup> <m:mi>μ</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>!</m:mo> </m:mrow> </m:mfrac> </m:mstyle> <m:mspace width="1em"/> </m:mtd> <m:mtd> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>k</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mn>0</m:mn> <m:mspace width="1em"/> </m:mtd> <m:mtd> <m:mo stretchy="false">(</m:mo> <m:mi>k</m:mi> <m:mo>></m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mfenced> </m:math> $$p^{\mu}_{nk}= \begin{cases} \dfrac{n!}{\Gamma(n+1,\mu)}\dfrac{\textrm{e}^{-\mu}\mu^k}{k!} \quad &(0\leq k\leq n), \\[1ex] 0\quad & (k>n), \end{cases}$$ where μ > 0 is fixed. We introduce the sequence space <jats:inline-graphic xmlns:xlink="htt