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An Analog of the Cauchy-Hadamard Formula for Expansions in q-Polynomials

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Book cover Theory and Applications of Special Functions

Part of the book series: Developments in Mathematics ((DEVM,volume 13))

Abstract

We derive an analog of the Cauchy-Hadamard formula for certain polynomial expansions and consider some examples.

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References

  • Ahlfors, L. V. (1979). Complex Analysis. McGraw-Hill, New York, third edition.

    Google Scholar 

  • Andrews, G. E. and Askey, R. A. (1985). Classical orthogonal polynomials. In Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), volume 1171 of Lecture Notes in Math., pages 36–62. Springer-Verlag, Berlin.

    Google Scholar 

  • Andrews, G. E., Askey, R. A., and Roy, R. (1999). Special Functions. Cambridge University Press, Cambridge.

    Google Scholar 

  • Antonov, V. A. and Kholshevnikov, K. V. (1979). An estimate for Jacobi polynomials in the complex domain. Vestnik Leningrad. Univ. Mat. Mekh. Astronom., (2):10–13, 118.

    MathSciNet  Google Scholar 

  • Askey, R. A. (1975). Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conferences Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA.

    Google Scholar 

  • Askey, R. A. and Wilson, J. A. (1985). Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc., 54(319).

    Google Scholar 

  • Atakishiyev, N. M. and Suslov, S. K. (1992). Difference hypergeometric functions. In Gonchar, A. A. and Saff, E. B., editors, Progress in Approximation Theory (Tampa, FL, 1990), volume 19 of Computational Mathematics, pages 1–35. Springer-Verlag, New York.

    Google Scholar 

  • Boas, R. P. and Buck, R. C. (1964). Polynomial Expansions of Analytic Functions. Springer-Verlag, Berlin. Second printing, corrected. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Bd. 19.

    Google Scholar 

  • Bromwich, T. J. (1965). An Introduction to the Theory of Infinite Series. Macmillan, New York. Second Edition, revised.

    Google Scholar 

  • Bustoz, J. and Suslov, S. K. (1998). Basic analog of Fourier series on a q-quadratic grid. Methods Appl. Anal., 5:1–38.

    MathSciNet  Google Scholar 

  • Carlson, B. C. (1974a). Expansions of analytic functions in Jacobi series. SIAM J. Math. Anal., 5:586–596.

    Article  MATH  MathSciNet  Google Scholar 

  • Carlson, B. C. (1974b). Inequalities for Jacobi polynomials and Dirichlet averages. SIAM J. Math. Anal., 5:797–808.

    Article  MATH  MathSciNet  Google Scholar 

  • Davis, P. J. (1963). Interpolation and Approximation. Blaisdell Publishing Co. Ginn and Co., New York-Toronto-London.

    Google Scholar 

  • Dienes, P. (1957). The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable. Dover, New York.

    Google Scholar 

  • Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). Higher Transcendental Functions, Vol. II. McGraw-Hill, New York-Toronto-London. Based, in part, on notes left by Harry Bateman.

    Google Scholar 

  • Floreanini, R. and Vinet, L. (1995). A model for the continuous q-ultraspherical polynomials. J. Math. Phys., 36:3800–3813.

    Article  MathSciNet  Google Scholar 

  • Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Cambridge University Press.

    Google Scholar 

  • Ismail, M. E. H., Rahman, M., and Stanton, D. (1999). Quadratic q-exponentials and connection coefficient problems. Proc. Amer. Math. Soc., 127(10):2931–2941.

    Article  MathSciNet  Google Scholar 

  • Ismail, M. E. H., Rahman, M., and Zhang, R. (1996). Diagonalization of certain integral operators II. J. Comp. Appl. Math., 68:163–196.

    Article  MathSciNet  Google Scholar 

  • Ismail, M. E. H. and Stanton, D. (2000). Addition theorems for the q-exponential functions. In Ismail, M. E. H. and Stanton, D. W., editors, q-Series from a Contemporary Perspective, volume 254 of Contemporary Mathematics, pages 235–245. American Mathematical Society, Providence, RI.

    Google Scholar 

  • Ismail, M. E. H. and Stanton, D. (2002). q-integral and moment representations for q-orthogonal polynomials. Canad. J. Math., 54(4):709–735.

    MathSciNet  Google Scholar 

  • Ismail, M. E. H. and Stanton, D. (2003a). Applications of q-Taylor theorems. J. Comp. Appl. Math., 153:259–272.

    Article  MathSciNet  Google Scholar 

  • Ismail, M. E. H. and Stanton, D. (2003b). q-Taylor theorems, polynomial expansions, and interpolation of entire functions. J. Approx. Theory, 123(1):125–146.

    Article  MathSciNet  Google Scholar 

  • Ismail, M. E. H. and Wilson, J. A. (1982). Asymptotic and generating relations for the q-Jacobi and 4ϕ3 polynomials. J. Approx. Theory, 36:43–54.

    Article  MathSciNet  Google Scholar 

  • Ismail, M. E. H. and Zhang, R. (1994). Diagonalization of certain integral operators. Advances in Math., 109:1–33.

    Article  MathSciNet  Google Scholar 

  • Kac, V. and Cheung, P. (2002). Quantum Calculus. Universitext. Springer-Verlag, New York.

    Google Scholar 

  • Markushevich, A. I. (1985). Theory of Functions of a Complex Variable, volume I. Chelsea Publishing Company, New York, second edition.

    Google Scholar 

  • Rahman, M. (1986). q-Wilson functions of the second kind. SIAM J. Math. Anal., 17:1280–1286.

    Article  MATH  MathSciNet  Google Scholar 

  • Rainville, E. D. (1960). Special Functions. The Macmillan Company, New York.

    Google Scholar 

  • Suslov, S. K. (1997). “Addition” theorems for some q-exponential and q-trigonometric functions. Methods Appl. Anal., 4:11–32.

    MATH  MathSciNet  Google Scholar 

  • Suslov, S. K. (2000). Another addition theorem for the q-exponential function. J. Phys. A: Math. Gen., 33:L375–380.

    Article  MATH  MathSciNet  Google Scholar 

  • Suslov, S. K. (2001). Basic exponential functions on a q-quadratic grid. In Bustoz, J., Ismail, M. E. H., and Suslov, S. K., editors, Special Functions 2000: Current Perspective and Future Directions, volume 30 of NATO Science Series II: Mathematics, Physics and Chemistry, pages 411–456. Kluwer Academic Publishers, Dordrecht-Boston-London.

    Google Scholar 

  • Suslov, S. K. (2002). Some expansions in basic Fourier series and related topics. J. Approx. Theory, 115(2):289–353.

    Article  MATH  MathSciNet  Google Scholar 

  • Suslov, S. K. (2003a). Expansion of analytic functions in q-orthogonal polynomials. Submitted.

    Google Scholar 

  • Suslov, S. K. (2003b). An Introduction to Basic Fourier Series, volume 9 of Developments in Mathematics. Kluwer Academic Publishers, Dordrecht-Boston-London.

    Google Scholar 

  • Szegő, G. (1975). Orthogonal Polynomials, volume XXIII of Amer. Math. Soc. Colloq. Publ. American Mathematical Society, Providence, RI, fourth edition.

    Google Scholar 

  • Szeghő, G. (1982). Collected Papers. Vol. 1. Contemporary Mathematicians. Birkhäuser, Boston, MA. 1915–1927, Edited by Richard Askey, Including commentaries and reviews by George Pólya, P. C. Rosenbloom, Askey, L. E. Payne, T. Kailath and Barry M. McCoy.

    Google Scholar 

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Suslov, S.K. (2005). An Analog of the Cauchy-Hadamard Formula for Expansions in q-Polynomials. In: Ismail, M.E., Koelink, E. (eds) Theory and Applications of Special Functions. Developments in Mathematics, vol 13. Springer, Boston, MA. https://doi.org/10.1007/0-387-24233-3_20

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