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Rogers–Szegő polynomials

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.2 Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) = H ( a , q n , m ; - n , β , γ , δ ; z )
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . These solutions are the Heun polynomials. …
2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
Normalization
Orthogonal Invariance
Summation
Mean-Value
3: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
4: 18.3 Definitions
§18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , see Abramowitz and Stegun (1964, pp. 793–801). … For another version of the discrete orthogonality property of the polynomials T n ( x ) see (3.11.9). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
5: 18.1 Notation
Classical OP’s
Hahn Class OP’s
Wilson Class OP’s
Nor do we consider the shifted Jacobi polynomials: …or the dilated Chebyshev polynomials of the first and second kinds: …
6: 18.33 Polynomials Orthogonal on the Unit Circle
§18.33 Polynomials Orthogonal on the Unit Circle
Askey
When a = 0 the Askey case is also known as the Rogers–Szegő case.
§18.33(v) Biorthogonal Polynomials on the Unit Circle
7: 17.18 Methods of Computation
Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …
8: David M. Bressoud
His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …
9: Bibliography
  • W. A. Al-Salam and L. Carlitz (1965) Some orthogonal q -polynomials. Math. Nachr. 30, pp. 47–61.
  • M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.
  • G. E. Andrews (1966b) q -identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33 (3), pp. 575–581.
  • G. E. Andrews (1984) Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.
  • R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
  • 10: 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.