Minimal solutions of three-term recurrence relations and orthogonal polynomials

Author:
Walter Gautschi

Journal:
Math. Comp. **36** (1981), 547-554

MSC:
Primary 33A65; Secondary 65D20, 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606512-6

MathSciNet review:
606512

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Abstract | References | Similar Articles | Additional Information

Abstract: We observe that the well-known recurrence relation for orthogonal polynomials admits a "minimal solution" if *z* is outside the spectrum of the mass distribution with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined. The minimal solution, indeed is , and can be computed accurately by means of the author's continued fraction algorithm. An application is made to special Gauss-type quadrature formulas.

**[1]**W. G. Bickley, L. J. Comrie, J. C. P. Miller, D. H. Sadler, and A. J. Thompson,*Bessel functions. Part II. Functions of positive integer order*, British Association for the Advancement of Science, Mathematical Tables, vol. X, University Press, Cambridge, 1952. MR**0050973****[2]**J. R. Cash,*An extension of Olver’s method for the numerical solution of linear recurrence relations*, Math. Comp.**32**(1978), no. 142, 497–510. MR**0483578**, https://doi.org/10.1090/S0025-5718-1978-0483578-8**[3]**J. R. Cash,*A note on Olver’s algorithm for the solution of second-order linear difference equations*, Math. Comp.**35**(1980), no. 151, 767–772. MR**572854**, https://doi.org/10.1090/S0025-5718-1980-0572854-5**[4]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[5]**P. Deuflhard,*A summation technique for minimal solutions of linear homogeneous difference equations*, Computing**18**(1977), no. 1, 1–13 (English, with German summary). MR**0433927**, https://doi.org/10.1007/BF02248773**[6]**J. D. Donaldson and David Elliott,*A unified approach to quadrature rules with asymptotic estimates of their remainders*, SIAM J. Numer. Anal.**9**(1972), 573–602. MR**0317522**, https://doi.org/10.1137/0709051**[7]**Walter Gautschi,*Computational aspects of three-term recurrence relations*, SIAM Rev.**9**(1967), 24–82. MR**0213062**, https://doi.org/10.1137/1009002**[8]**Walter Gautschi,*Questions of numerical conditions related to polynomials*, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 45–72. MR**519056****[9]**Walter Gautschi,*On generating Gaussian quadrature rules*, Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978), Internat. Ser. Numer. Math., vol. 45, Birkhäuser, Basel-Boston, Mass., 1979, pp. 147–154. MR**561288****[10]**Gene H. Golub and John H. Welsch,*Calculation of Gauss quadrature rules*, Math. Comp. 23 (1969), 221-230; addendum, ibid.**23**(1969), no. 106, loose microfiche suppl, A1–A10. MR**0245201**, https://doi.org/10.1090/S0025-5718-69-99647-1**[11]**Ravindra Kumar,*A class of quadrature formulas*, Math. Comp.**28**(1974), 769–778. MR**0373240**, https://doi.org/10.1090/S0025-5718-1974-0373240-0**[12]**R. Kumar,*Certain Gaussian quadratures*, J. Inst. Math. Appl.**14**(1974), 175–182. MR**0356452****[13]**J. M. Smith, F. W. J. Olver, and D. W. Lozier,*Extended-range arithmetic and normalized Legendre polynomials*, ACM Trans. Math. Software**7**(1981), no. 1, 93–105. MR**607353**, https://doi.org/10.1145/355934.355940**[14]**R. M. M. Mattheij and A. van der Sluis,*Error estimates for Miller’s algorithm*, Numer. Math.**26**(1976), no. 1, 61–78. MR**0438745**, https://doi.org/10.1007/BF01396566**[15]***Handbook of mathematical functions, with formulas, graphs and mathematical tables*, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966. MR**0208798****[16]**F. W. J. Olver,*Numerical solution of second-order linear difference equations*, J. Res. Nat. Bur. Standards Sect. B**71B**(1967), 111–129. MR**0221789****[17]**F. W. J. Olver and D. J. Sookne,*Note on backward recurrence algorithms*, Math. Comp.**26**(1972), 941–947. MR**0331826**, https://doi.org/10.1090/S0025-5718-1972-0331826-1**[18]**Oskar Perron,*Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche*, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957 (German). MR**0085349****[19]**Thomas E. Price Jr.,*Orthogonal polynomials for nonclassical weight functions*, SIAM J. Numer. Anal.**16**(1979), no. 6, 999–1006. MR**551321**, https://doi.org/10.1137/0716073**[20]**J. A. Shohat and J. D. Tamarkin,*The Problem of Moments*, American Mathematical Society Mathematical surveys, vol. I, American Mathematical Society, New York, 1943. MR**0008438****[21]**J. M. Smith, F. W. J. Olver, and D. W. Lozier,*Extended-range arithmetic and normalized Legendre polynomials*, ACM Trans. Math. Software**7**(1981), no. 1, 93–105. MR**607353**, https://doi.org/10.1145/355934.355940**[22]**Henry C. Thacher Jr.,*Series solutions to differential equations by backward recurrence*, Information processing 71 (Proc. IFIP Congress, Ljubljana, 1971) North-Holland, Amsterdam, 1972, pp. 1287–1291. MR**0461923****[23]**Henry C. Thacher Jr.,*New backward recurrences for Bessel functions*, Math. Comp.**33**(1979), no. 146, 744–764. MR**521289**, https://doi.org/10.1090/S0025-5718-1979-0521289-1**[24]**P. Van der Cruyssen,*A reformulation of Olver’s algorithm for the numerical solution of second-order linear difference equations*, Numer. Math.**32**(1979), no. 2, 159–166. MR**529906**, https://doi.org/10.1007/BF01404872**[25]**P. Van der Cruyssen,*Linear difference equations and generalized continued fractions*, Computing**22**(1979), no. 3, 269–278 (English, with German summary). MR**620219**, https://doi.org/10.1007/BF02243567**[26]**R. V. M. Zahar, "A mathematical analysis of Miller's algorithm,"*Numer. Math.*, v. 27, 1977, pp. 427-447.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606512-6

Keywords:
Minimal solutions of three-term recurrence relations,
orthogonal polynomials,
moment problem,
modified moments

Article copyright:
© Copyright 1981
American Mathematical Society