Exceptional AskeyWilson polynomials and continued fractions
Authors:
Dharma P. Gupta and David R. Masson
Journal:
Proc. Amer. Math. Soc. 112 (1991), 717727
MSC:
Primary 33D45; Secondary 39A10, 40A15
MathSciNet review:
1059625
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Abstract: Two linearly independent solutions of the threeterm recurrence relation for the AskeyWilson polynomials are obtained for the special cases .By obtaining the subdominant solution and employing Pincherle's theorem, the associated continued fractions and properties of the corresponding weight functions are derived. The cases are exceptional. They differ from the cases considered by Askey and Wilson [1 ] and are limits of a family of associated cases considered by Ismail and Rahman [5].
 [1]
Richard
Askey and James
Wilson, Some basic hypergeometric orthogonal polynomials that
generalize Jacobi polynomials, Mem. Amer. Math. Soc.
54 (1985), no. 319, iv+55. MR 783216
(87a:05023)
 [2]
T. J. I'A. Bromwich, An introduction to the theory of infinite series, 2nd ed., Macmillan, New York, 1955.
 [3]
Mourad
E. H. Ismail and James
A. Wilson, Asymptotic and generating relations for the
𝑞Jacobi and ₄𝜙₃ polynomials, J. Approx.
Theory 36 (1982), no. 1, 43–54. MR 673855
(84e:33012), http://dx.doi.org/10.1016/00219045(82)900697
 [4]
M.
E. H. Ismail, J.
Letessier, G.
Valent, and J.
Wimp, Two families of associated Wilson polynomials, Canad. J.
Math. 42 (1990), no. 4, 659–695. MR 1074229
(92b:33021), http://dx.doi.org/10.4153/CJM19900354
 [5]
Mourad
E. H. Ismail and Mizan
Rahman, The associated AskeyWilson
polynomials, Trans. Amer. Math. Soc.
328 (1991), no. 1,
201–237. MR 1013333
(92c:33019), http://dx.doi.org/10.1090/S00029947199110133334
 [6]
William
B. Jones and Wolfgang
J. Thron, Continued fractions, Encyclopedia of Mathematics and
its Applications, vol. 11, AddisonWesley Publishing Co., Reading,
Mass., 1980. Analytic theory and applications; With a foreword by Felix E.
Browder; With an introduction by Peter Henrici. MR 595864
(82c:30001)
 [7]
David
R. Masson, Difference equations, continued fractions, Jacobi
matrices and orthogonal polynomials, Nonlinear numerical methods and
rational approximation (Wilrijk, 1987), Math. Appl., vol. 43, Reidel,
Dordrecht, 1988, pp. 239–257. MR 1005362
(90i:33020)
 [8]
David
R. Masson, Wilson polynomials and some continued fractions of
Ramanujan, Proceedings of the U.S.Western Europe Regional Conference
on Padé Approximants and Related Topics (Boulder, CO, 1988), 1991,
pp. 489–499. MR 1113939
(92h:33015), http://dx.doi.org/10.1216/rmjm/1181073019
 [9]
, Associated Wilson polynomials, Constructive Approximation (to appear).
 [10]
David
R. Masson, Difference equations revisited, Mathematical
quantum field theory and related topics (Montreal, PQ, 1987), CMS Conf.
Proc., vol. 9, Amer. Math. Soc., Providence, RI, 1988,
pp. 73–82. MR 973460
(89m:39005)
 [11]
D.
B. Sears, On the transformation theory of basic hypergeometric
functions, Proc. London Math. Soc. (2) 53 (1951),
158–180. MR 0041981
(13,33d)
 [12]
G. Szegö, Orthogonal polynomials, Amer. Math. Soc., Providence, RI, 1975.
 [13]
J.
A. Wilson, Threeterm contiguous relations and some new orthogonal
polynomials, Padé and rational approximation (Proc. Internat.
Sympos., Univ. South Florida, Tampa, Fla., 1976), Academic Press, New
York, 1977, pp. 227–232. MR 0466671
(57 #6548)
 [14]
J. Wilson, Hypergeometric series, recurrence relations and some orthogonal functions, Ph.D. diss., University of Wisconsin, Madison, 1978.
 [1]
 R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., No. 319, Amer. Math. Soc., Providence, RI, 1985, pp. 155. MR 783216 (87a:05023)
 [2]
 T. J. I'A. Bromwich, An introduction to the theory of infinite series, 2nd ed., Macmillan, New York, 1955.
 [3]
 M. E. H. Ismail and J. A. Wilson, Asymptotic and generating relations for the Jacobi and polynomials, J. Approx. Theory 36 (1982), 4354. MR 673855 (84e:33012)
 [4]
 M. E. H. Ismail, J. Letessier, G. Valent, and J. Wimp, Two families of associated Wilson polynomials, Canad. J. Math. 42 (1990), 659695. MR 1074229 (92b:33021)
 [5]
 M. E. H. Ismail and M. Rahman, The associated AskeyWilson polynomials, Trans. Amer. Math. Soc. (to appear). MR 1013333 (92c:33019)
 [6]
 W. B. Jones and W. J. Thron, Continued fractions: Analytic theory and applications, AddisonWesley, Reading, MA, 1980. MR 595864 (82c:30001)
 [7]
 D. R. Masson, Difference equations, continued fractions, Jacobi matrices and orthogonal polynomials, in Nonlinear numerical methods and rational approximation (A. Cuyt, ed.), Reidel, Dordrecht, 1988, pp. 239257. MR 1005362 (90i:33020)
 [8]
 , Wilson polynomials and some continued fractions of Ramanujan, Rocky Mountain J. Math. (to appear). MR 1113939 (92h:33015)
 [9]
 , Associated Wilson polynomials, Constructive Approximation (to appear).
 [10]
 , Difference equations revisited, Canad. Math. Soc. Conf. Proc. (J. S. Feldman and L. M. Rosen, eds.), vol. 9, Amer. Math. Soc., Providence, RI, 1988, pp. 7382. MR 973460 (89m:39005)
 [11]
 D. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53 (1951), 158180. MR 0041981 (13:33d)
 [12]
 G. Szegö, Orthogonal polynomials, Amer. Math. Soc., Providence, RI, 1975.
 [13]
 J. A. Wilson, Threeterm contiguous relations and some new orthogonal polynomials, Padé and Rational Approximation (Proc. Internat. Sympos. Univ. South Florida, 1976), Academic Press, New York, 1977, pp. 227232. MR 0466671 (57:6548)
 [14]
 J. Wilson, Hypergeometric series, recurrence relations and some orthogonal functions, Ph.D. diss., University of Wisconsin, Madison, 1978.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919911059625X
PII:
S 00029939(1991)1059625X
Keywords:
AskeyWilson polynomials,
threeterm recurrence,
subdominant solution,
Pincherle's theorem,
continued fractions,
weight functions,
mass points
Article copyright:
© Copyright 1991 American Mathematical Society
