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Transformation of certain bilinear generating functions

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Summary

Erdélyi's generalizations of the Hardy-Hille formula are extended to series involving arbitrary coefficients; see(1.6), …,(1.9) below. These identities may be thought of as identities in formal power series. The proofs are simple: each identity in shown to be equivalent to an algebraic identity that is easily verified.

References

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    N. Abdul-Halim andW. A. Al-Salam Double Euler trasformations of certain hypergeometric functions, Duke Mathematical Journal, vol. 30 (1963), pp. 51–62.

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    H. Buckholtz,Die Konfluente Hypergeometrische Funktionen, Berlin 1953.

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    A. Erdélyi,Transformation einer gewissen nach Produkten konfluenten hypergeometrischer Funktionen fortschreitenden Reihe, Compositio Mathematica, vol. 6 (1939), pp. 336–347.

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    —— ——,Transformation of a certain series of products of confluent hypergeometric functions, Applications to Laguerre and Charlier polynomials, Compositio Mathematica, vol. 7 (1940), pp. 340–352.

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    E. D. Rainville,Special Functions, New York, (1960).

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    L. Weisner,Group-theoretic orgin of certain generating functions, Pacific Journal of Mathematics, vol. 5 (1955), pp. 1033–1039.

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Supported in part by NSF grant GP-7855.

Entrata in Redazione il 18 novembre 1969.

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Carlitz, L. Transformation of certain bilinear generating functions. Annali di Matematica 86, 155–168 (1970). https://doi.org/10.1007/BF02415716

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Keywords

  • Generate Function
  • Power Series
  • Formal Power Series
  • Algebraic Identity
  • Arbitrary Coefficient