Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-20T00:55:23.053Z Has data issue: false hasContentIssue false
  • English
  • Français

On a dual relation for addition formulas of additive groups II

Published online by Cambridge University Press:  22 January 2016

Toshihiro Watanabe
Toshihiro Watanabe*
Affiliation:
Department of Applied Mathematics, Faculty of Engineering, Gifu University, Gifu 500, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is a continuation of our previous memoir [28], hereafter referred to as I, and constitutes the second chapter of this series. As stated in I, our aim in this series is to examine properties of a polynomial sequence with several variables satisfying an addition formula by means of the down-ladder, and to give a generalization of so called classical polynomials.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 97 , March 1985 , pp. 95 - 135
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

[ 1 ] Appell, P., Sur une classe de polynômes, Ann. Sci. École Norm. Sup., 9 (1880), 119–144.Google Scholar
[ 2 ] Appell, P. and Kampé de Fériet, J., Fonctions Hypergéométriqués et Hypersheriques, Polynômes d’Hermite, Gauthier-Villars, Paris, 1926.Google Scholar
[ 3 ] Bell, E. T., The history of Blissard symbolic method with a sketch of the inventor’s life, Amer. Math. Monthly, XLV (1938), 414–421.Google Scholar
[ 4 ] Bell, E. T., Postulational bases for the umbral calculas, Amer. J. Math., 62 (1940), 717–724.CrossRefGoogle Scholar
[ 5 ] Blissard, J., Theory of generic equations, Quart. J. Pure Appl. Math., 4 (1861), 279–305; 5 (1862), 58–75, 184–208.Google Scholar
[ 6 ] Boas, R. P. and Buck, R. C., Polynomial Expansions of Analytic Functions, Springer, Berlin, 1964.Google Scholar
[ 7 ] Brown, J. W., On zero type sets of Laguerre polynomials, Duke Math. J., 35 (1968), 821–823.Google Scholar
[ 8 ] Burchnall, J. L., A note on the polynomials of Hermite, Quart. J. Math., 12 (1941), 9–11.Google Scholar
[ 9 ] Carlitz, L., Some generating functions for Laguerre polynomials, Duke Math. J., 35 (1968), 825–827.Google Scholar
[10] Carlitz, L., A class of generating functions, SIAM J. Math. Anal., 8, no. 3, (1977), 517–532.CrossRefGoogle Scholar
[11] Erdélyi, A. ed., Higher Transcendental Functions, vol. I, vol. II, vol. III, McGraw-Hill, New York, 1953.Google Scholar
[12] Feldheim, E., Quelques nouvelles relations pour les polynomes d’Hermite, J. Lond. Math. Soc, 13 (1938), 22–29.Google Scholar
[13] Feldheim, E., Équations integrales pour les polynomes d’Hermite á une et plusieurs variables, pour les polynomes de Laguerre, et pour les fonctions hypergéométriques les plus generales, Ann. Scoula Norm. Super. Pisa, (2) 9 (1940), 225–252.Google Scholar
[14] Grace, J. H. and Young, A., The Algebra of Invariant, Chelsea, New York, reprinted from 1903.Google Scholar
[15] Jordan, K., Calculus of Finite Differences, Chelsea, New York, 1965, (reprinted from 1947).Google Scholar
[16] Lah, I., Eine neue art von zahlen, ihre eigenshaften und anwendung in der mathe matischen statistik, Mitteilungsbl. Math. Statist., 7 (1955), 203–216.Google Scholar
[17] Lagrange, R., Memoire sur les suites de polynomes, Acta Math., 51 (1928), 201–309.Google Scholar
[18] Nörlund, N. E., Vorlesungen ueber Differenzen Rechnung, Chelsea, New York, 1954 (reprinted from 1928).Google Scholar
[19] Nielsen, N., Traité Elémentaire des Nombres de Bernoulli, Gauthier-Villas, Paris, 1923.Google Scholar
[20] Matsuda, M., The Theory of Exterior Differential Forms, Iwanami, Tokyo, 1975, (in Japanese).Google Scholar
[21] Rota, G.C., The number of partitions of a set, Amer. Math. Monthly, 71 (1964), 498–504.Google Scholar
[22] Rota, G. C, Kahaner, D. and Odlyzko, A., Finite operator calculus, J. Math. Anal. Appl., 42 (1973), 685–760.CrossRefGoogle Scholar
[23] Sheffer, I. M., A differential equation for Appell polynomials, Bull. Amer. Math. Soc, 41 (1935), 914–923.Google Scholar
[24] Sheffer, I. M., Some properties of polynomial sets of type zero, Duke Math. J., 5 (1939), 590–622.Google Scholar
[25] Srivastava, H. M. and Singhal, J. P., A unified presentation of certain classical polynomials, Math. Comp., 26, no. 120, (1972), 965–975.CrossRefGoogle Scholar
[26] Steffensen, J. E., The poweroid, an extension of the mathematical notion of power, Acta Math., 73 (1941), 333–366.CrossRefGoogle Scholar
[27] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc, New York, 1959.Google Scholar
[28] Watanabe, T., On a dual relation for addition formulas of additive groups I, Nagoya Math. J., 94 (1984), 171–191.Google Scholar
[29] Watson, G. N., A note of the polynomials of Hermite and Laguerre, J. Lond. Math. Soc, 13 (1938), 29–32.Google Scholar
[30] Watson, G. N., Notes on generating functions of polynomials: Hermite polynomials, J. Lond. Math. Soc, 8 (1933), 194–199.Google Scholar