Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-14T07:01:23.519Z Has data issue: false hasContentIssue false
  • English
  • Français

A recurrence relation generalizing those of Apéry

Published online by Cambridge University Press:  09 April 2009

Richard Askey  and
J. A. Wilson
Richard Askey
Affiliation:
Department of Mathematics University of Wisconsin-MadisonVan Vleck Hall Madison, Wisconsin 53706, U.S.A.
J. A. Wilson
Affiliation:
Department of Mathematics Iowa State University Ames, Iowa 50011, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

A three term recurrence relation is found for

when a + d = b + c. This includes the recurrence relations of Apéry associated with ζ(3), ζ(2) and log 2 as special or limiting cases.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 2 , April 1984 , pp. 267 - 278
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Askey, R. and Gasper, G., ‘Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients,’ Proc. Cambridge Philos. Soc. 70 (1971), 243–255.CrossRefGoogle Scholar
[2]Askey, R. and Wilson, J., ‘A set of orthogonal polynomials that generalize the Racah coefficients or 6 – j symbols,’ SIAM J. Math. Anal. 10 (1979), 1008–1016.CrossRefGoogle Scholar
[3]Gasper, G., ‘Linearization of the product of Jacobi polynomials, II,’ Canad. J. Math. 22 (1970), 582–593.CrossRefGoogle Scholar
[4]Gauss, C. F., ‘Disquisitiones generales circa seriem infinitam…,’ Comment Gotting. 2 (1812), 1–46; Werke, III (1868), 123–162.Google Scholar
[5]Kummer, E. E., ‘Über die hypergeometrische Reihe,’ J. für Math. 15 (1836), 39–83, 127–172;Google Scholar
Collected Papers, II, 75–166.Google Scholar
[6]Mendes-France, M., Roger Apéry et l'irrationnel, (Le Recherche, No. 97).Google Scholar
[7]Pfaff, J. F., ‘Observations analyticae ad L. Euler's Institutiones Calculi Integralis,’ vol. IV, Supplem. II et IV, Historte de 1793, Nova acta academiae scientiarum Petropolitanae, XI, 1797, 38–57. (Note, the history section is paged separately from the scientific section of this journal.)Google Scholar
[8]Rainville, E. D., Special functions (Macmillan, New York, 1960).Google Scholar
[9]Rahman, M., ‘A non-negative representation of the linearization coefficients of the product of Jacobi polynomials,’ Canad. J. Math. 33 (1981), 915–928.CrossRefGoogle Scholar
[10]Raynal, J., ‘On the definition and properties of generalized 3 — j symbols,’ J. Math. Phys. 19 (1978), 467–476.CrossRefGoogle Scholar
[11]Raynal, J., ‘On the definition annd properties of generalized 6 – j symbols,’ J. Math. Phys. 20 (1979), 2398–2415.CrossRefGoogle Scholar
[12]Rieger, G. J., ‘Einige Rekursionsformeln für Summen mit Binomialkoeffizienten,’ Abh. Braunschweig. Wiss. Gesellsch 31 (1980), 137–143.Google Scholar
[13]Szegö, G., Orthogonal polynomials, fourth edition (Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, R. I., 1975).Google Scholar
[14]van der Poorten, A., ‘A proof that Euler missed…Apéry's proof of the irrationality of ζ(3),’ Math. Intelligencer 1 (1979), 195–203.CrossRefGoogle Scholar
[15]Whipple, F. J. W., ‘On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum,’ Proc. London Math. Soc. (2) 24 (1926), 247–263.CrossRefGoogle Scholar
[16]Whipple, F. J. W., ‘Some transformations of generalized hypergeometric series,’ Proc. London Math. Soc. (2) 26 (1927), 257–272.CrossRefGoogle Scholar
[17]Wilson, J. A., ‘Three-term contiguous relations and some new orthogonal polynomials,’ Padé and rational approximation, edited by Saff, E. B. and Varga, R. S. (Academic Press, New York, 1977, 227–232).CrossRefGoogle Scholar
[18]Wilson, J. A., Hypergeometric series recurrence relations and some new orthogonal functions, (Ph.D. thesis, Univ. of Wisconsin-Madison, Madison, 1978).Google Scholar
[19]Wilson, J. A., ‘Some hypergeometric orthogonal polynomials,’ SIAM J. Math. Anal. 11 (1980), 690–701.CrossRefGoogle Scholar