The parameters of a chain sequence

Author:
T. S. Chihara

Journal:
Proc. Amer. Math. Soc. **108** (1990), 775-780

MSC:
Primary 40A15; Secondary 42C05

MathSciNet review:
1002153

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a method for constructing explicitly all parameter sequences for any chain sequence for which one parameter sequence is known. An application to orthogonal polynomials associated with birth and death processes is given.

**[1]**T. S. Chihara,*Chain sequences and orthogonal polynomials*, Trans. Amer. Math. Soc.**104**(1962), 1–16. MR**0138933**, 10.1090/S0002-9947-1962-0138933-7**[2]**T. S. Chihara,*An introduction to orthogonal polynomials*, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR**0481884****[3]**T. S. Chihara,*Spectral properties of orthogonal polynomials on unbounded sets*, Trans. Amer. Math. Soc.**270**(1982), no. 2, 623–639. MR**645334**, 10.1090/S0002-9947-1982-0645334-4**[4]**T. S. Chihara,*Orthogonal polynomials and measures with end point masses*, Rocky Mountain J. Math.**15**(1985), no. 3, 705–719. MR**813269**, 10.1216/RMJ-1985-15-3-705**[5]**T. S. Chihara,*Hamburger moment problems and orthogonal polynomials*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 189–203. MR**986686**, 10.1090/S0002-9947-1989-0986686-1**[6]**Mourad E. H. Ismail,*Monotonicity of zeros of orthogonal polynomials*, 𝑞-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 177–190. MR**1019851**, 10.1007/978-1-4684-0637-5_14**[7]**L. Jacobsen and D. Masson,*On the convergence of limit periodic continued fractions*,*where*, part III, Constructive Approx., (to appear).**[8]**S. Karlin and J. L. McGregor,*The differential equations of birth-and-death processes, and the Stieltjes moment problem*, Trans. Amer. Math. Soc.**85**(1957), 489–546. MR**0091566**, 10.1090/S0002-9947-1957-0091566-1**[9]**E. P. Merkes,*On truncation errors for continued fraction computations*, SIAM J. Numer. Anal.**3**(1966), 486–496. MR**0202283****[10]**Paul G. Nevai,*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, 10.1090/memo/0213**[11]**V. B. Uvarov,*The connection between systems of polynomials that are orthogonal with respect to different distribution functions*, Ž. Vyčisl. Mat. i Mat. Fiz.**9**(1969), 1253–1262 (Russian). MR**0262764****[12]**Erik A. van Doorn,*Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices*, J. Approx. Theory**51**(1987), no. 3, 254–266. MR**913621**, 10.1016/0021-9045(87)90038-4**[13]**H. S. Wall,*Analytic Theory of Continued Fractions*, D. Van Nostrand Company, Inc., New York, N. Y., 1948. MR**0025596**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
40A15,
42C05

Retrieve articles in all journals with MSC: 40A15, 42C05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1002153-7

Keywords:
Chain sequence,
continued fractions,
orthogonal polynomials,
birth and death process

Article copyright:
© Copyright 1990
American Mathematical Society