About the Project
NIST

Rogers polynomials

AdvancedHelp

(0.002 seconds)

10 matching pages

1: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
2: 18.33 Polynomials Orthogonal on the Unit Circle
3: 18.28 Askey–Wilson Class
These polynomials are also called Rogers polynomials. …
4: Bibliography B
  • H. Bateman (1905) A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.
  • G. Baxter (1961) Polynomials defined by a difference system. J. Math. Anal. Appl. 2 (2), pp. 223–263.
  • R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.
  • S. L. Belousov (1962) Tables of Normalized Associated Legendre Polynomials. Pergamon Press, The Macmillan Co., Oxford-New York.
  • A. Berkovich and B. M. McCoy (1998) Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.
  • 5: 10.23 Sums
    where C k ( ν ) ( cos α ) is Gegenbauer’s polynomial18.3). … For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). … and O k ( t ) is Neumann’s polynomial, defined by the generating function:
    10.23.12 1 t - z = J 0 ( z ) O 0 ( t ) + 2 k = 1 J k ( z ) O k ( t ) , | z | < | t | .
    O n ( t ) is a polynomial of degree n + 1 in 1 / t : O 0 ( t ) = 1 / t and …
    6: Bibliography L
  • D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.
  • D. H. Lehmer (1940) On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.
  • J. Lepowsky and R. L. Wilson (1982) A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.
  • J. L. López and N. M. Temme (1999a) Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.
  • J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.
  • 7: Bibliography R
  • M. Rahman (2001) The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.
  • D. St. P. Richards (Ed.) (1992) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.
  • M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
  • 8: 17.2 Calculus
    17.2.27 [ n m ] q = ( q ; q ) n ( q ; q ) m ( q ; q ) n - m = ( q - n ; q ) m ( - 1 ) m q n m - ( m 2 ) ( q ; q ) m ,
    17.2.30 [ - n m ] q = [ m + n - 1 m ] q ( - 1 ) m q - m n - ( m 2 ) ,
    §17.2(vi) Rogers–Ramanujan Identities
    9: 16.4 Argument Unity
    Rogers–Dougall Very Well-Poised Sum
    The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs. … One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … …
    10: 26.10 Integer Partitions: Other Restrictions
    26.10.3 ( 1 - x ) m , n = 0 p m ( k , 𝒟 , n ) x m q n = m = 0 k [ k m ] q q m ( m + 1 ) / 2 x m = j = 1 k ( 1 + x q j ) , | x | < 1 ,
    §26.10(iv) Identities
    Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …