Abstract
Delsarte’s approach to generalized translation operators via the solution of an associated Cauchy problem is used to derive the product formulas for the Bessel, Whittaker, and Jacobi functions in kernel form. As essential prerequisites, explicit representations of the corresponding Riemann functions are given for three cases. The main part of the paper deals with the Jacobi case, for which the derivation of the translation kernel is carried out explicitly. In the Bessel case, the results of Delsarte are covered, and in the Whittaker case, a generalization of previous results of the author on the Laguerre polynomial product formula is obtained.
Supported by the Deutsche Forschungsgemeinschaft under grant No. Go 261/5–2.
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References
Braaksma, B.L.J. — de Snoo, H.S.V., Generalized translation operators associated with a singular differential operator. In: Theory of Ordinary and Partial Differential Equations (B.D. Sleeman, I.M. Michaels, eds.) Lecture Notes Math. 415, Springer, Berlin 1974, 62–77.
Chaundy, T.W., Linear partial differential equations (I). Quart. J. Math. Oxford Ser. 9 (1938), 234–240.
Delsarte, J., Sur une extension de la formule de Taylor. J. Math. Pures Appl. 17 (1936), 213–231.
Erdelyi, A., et al., Higher Transcendental Functions, Vol. 1. McGraw Hill, New York 1953.
Flensted-Jensen, M. — Koornwinder, T.H., The convolution structure for Jacobi function expansions. Ark. Mat. 11 (1973), 245–262.
Glaeske, H.-J., Die Laguerre — Pinney — Transformation. Aequationes Math. 22 (1981), 73–85.
Henrici, P., A survey of I.N. Vekua’s theory of elliptic differential equations with analytic coefficients. Z. Angew. Math. Phys. 8(1957), 169–203.
Koornwinder, T.H., The addition formula for Jacobi polynomials, I. Summary of results. Indag. Math. 34 (1972), 188–191.
Markett, C., Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter. Anal. Math. 8 (1982), 19–37.
Markett, C., A new proof of Watson’s product formula for Laguerre polynomials via a Cauchy problem associated with a singular differential operator. (to appear)
Markett, C., Norm estimates for generalized translation operators associated with a singular differential operator. (to appear)
Riemann, B., Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Gesammelte Math. Werke, Leipzig 1892, 156–181.
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Markett, C. (1984). Product Formulas for Bessel, Whittaker, and Jacobi Functions via the Solution of an Associated Cauchy Problem. In: Butzer, P.L., Stens, R.L., Sz.-Nagy, B. (eds) Anniversary Volume on Approximation Theory and Functional Analysis. ISNM 65: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 65. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5432-0_39
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DOI: https://doi.org/10.1007/978-3-0348-5432-0_39
Publisher Name: Birkhäuser, Basel
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