An identity on pairs of Appell-type polynomials

TL;DRAbstract

In this paper, we define a sequence of polynomials <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:math> depending only on the choice of two analytic functions A and H in a neighborhood of zero. For a pair of compositional inverses A and B , we will show the identity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="true">)</mm

Chat with Paper

AI Agents for this Paper