From Wikipedia, the free encyclopedia

In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw ( 2010, 14) give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by

Recurrence and difference relations

with the initial conditions

From the above, one can easily calculate:

Generating function

where .

References

  • Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN  978-0-521-83357-8, MR  2128719
  • Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi: 10.1007/978-3-642-05014-5, ISBN  978-3-642-05013-8, MR  2656096
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN  978-0-521-19225-5, MR  2723248.
  • Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX  10.1.1.643.3896.
From Wikipedia, the free encyclopedia

In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw ( 2010, 14) give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by

Recurrence and difference relations

with the initial conditions

From the above, one can easily calculate:

Generating function

where .

References

  • Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN  978-0-521-83357-8, MR  2128719
  • Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi: 10.1007/978-3-642-05014-5, ISBN  978-3-642-05013-8, MR  2656096
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN  978-0-521-19225-5, MR  2723248.
  • Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX  10.1.1.643.3896.

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