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THE POWER COLLECTION METHOD FOR CONNECTION RELATIONS: MEIXNER POLYNOMIALS.

Michael A Baeder1, Howard S Cohl2, Roberto S Costas-Santos3

  • 1Institutional Clients Group, Citigroup Inc., New York, NY 10013, USA.

Journal of Classical Analysis
|December 29, 2022
PubMed
Summary
This summary is machine-generated.

A new power collection method simplifies deriving connection relations for hypergeometric orthogonal polynomials. This technique yields generalized generating functions and integral expressions for Meixner and Krawtchouk polynomials.

Keywords:
05A1530E2033C2033C4534L10Generating functionsconnection coefficientsconnection-type relationsdefinite integralseigenfunction expansionsinfinite series

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Area of Science:

  • Mathematical Physics
  • Special Functions
  • Orthogonal Polynomials

Background:

  • The (q-)Askey scheme classifies hypergeometric orthogonal polynomials.
  • Deriving connection relations is crucial for understanding polynomial properties.
  • Existing methods can be complex and limited in scope.

Purpose of the Study:

  • Introduce a novel "power collection method" for deriving connection relations.
  • Demonstrate the method's applicability to Meixner and Krawtchouk polynomials.
  • Derive generalized generating functions and their integral representations.

Main Methods:

  • The core method involves "power collection" to establish polynomial connections.
  • Applied to Meixner and Krawtchouk polynomials within the (q-)Askey scheme.
  • Utilized orthogonality to derive contour integral and infinite series expressions.

Main Results:

  • Successfully derived connection and connection-type relations for Meixner and Krawtchouk polynomials.
  • Developed generalized generating functions with coefficients expressed via multiple hypergeometric functions.
  • Obtained corresponding contour integral and infinite series representations.

Conclusions:

  • The power collection method offers an efficient approach for analyzing hypergeometric orthogonal polynomials.
  • The derived generalized generating functions and integral forms expand the toolkit for special function research.
  • This work provides a unified framework for certain polynomial families in the (q-)Askey scheme.