Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions
Authors: Alfredo Deaño and Javier Segura
Journal: Math. Comp. 76 (2007), 879-901
MSC (2000): Primary 33C05, 33C15, 39A11, 40A15, 65D20
Published electronically: January 10, 2007
MathSciNet review: 2291841
Full-text PDF Free Access
Abstract: Three term recurrence relations can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their role for finite values of with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal. This phenomenon, related to Gautschi's anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions, is shown to be a general situation which takes place for recurrences with negative and changing sign once. We analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations (modified Bessel functions, confluent and Gauss hypergeometric functions) and discuss the implication of such transitory behaviour on the numerical stability of recursion.
- 1. Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
- 2. Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, https://doi.org/10.1137/1009002
- 3. Walter Gautschi, Anomalous convergence of a continued fraction for ratios of Kummer functions, Math. Comp. 31 (1977), no. 140, 994–999. MR 442204, https://doi.org/10.1090/S0025-5718-1977-0442204-3
- 4. A. Gil, J. Segura, N.M. Temme. Numerically satisfactory solutions of hypergeometric recursions. Accepted for publication in Math. Comput.
- 5. A. Gil, J. Segura, N.M. Temme. Numerical methods for special functions. SIAM (2006).
- 6. William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
- 7. Nico M. Temme, Recent problems from uniform asymptotic analysis of integrals in particular in connection with Tricomi’s Ψ-function, Tricomi’s ideas and contemporary applied mathematics (Rome/Turin, 1997) Atti Convegni Lincei, vol. 147, Accad. Naz. Lincei, Rome, 1998, pp. 183–201. MR 1737496
- 8. N. M. Temme, The numerical computation of the confluent hypergeometric function 𝑈(𝑎,𝑏,𝑧), Numer. Math. 41 (1983), no. 1, 63–82. MR 696551, https://doi.org/10.1007/BF01396306
- M. Abramowitz, I. Stegun (Eds). Handbook of Mathematical Functions with formulas, graphs, and mathematical tables. National Bureau of Standards. Applied Mathematics Series, no. 55. U.S. Government Printing Office, Washington DC (1964).MR 0167642
- W. Gautschi. Computational aspects of three-term recurrence relations. SIAM Review 9, no.1 (1967) 24-82. MR 0213062
- W. Gautschi. Anomalous convergence of a continued fraction for ratios of Kummer functions. Math. Comput., 31, no.140 (1977) 994-999. MR 0442204
- A. Gil, J. Segura, N.M. Temme. Numerically satisfactory solutions of hypergeometric recursions. Accepted for publication in Math. Comput.
- A. Gil, J. Segura, N.M. Temme. Numerical methods for special functions. SIAM (2006).
- W.B. Jones and W.J. Thron. Continued fractions. Analytic theory and applications. Encyclopaedia of Mathematics and its applications. Addison-Wesley Publishing Co., Reading, Mass., 1980. MR 0595864
- N.M. Temme. Recent problems from uniform asymptotic analysis of integrals, in particular in connection with Tricomi's psi-function. Tricomi's ideas and contemporary applied mathematics (Rome/Turin, 1997), 183-201, Atti Convegni Lincei, 147. MR 1737496 (2001i:33004)
- N.M. Temme. The numerical computation of the confluent hypergeometric function . Numer. Math. 41 (1983) 63-82. MR 0696551
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911-Leganés (Madrid), Spain
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005-Santander, Spain
Keywords: Hypergeometric functions, recurrence relations, condition and stability, continued fractions, numerical evaluation of special functions
Received by editor(s): February 1, 2006
Received by editor(s) in revised form: March 24, 2006
Published electronically: January 10, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.