type 2 Pollaczek polynomials counterexample
(0.001 seconds)
1—10 of 845 matching pages
1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
►Let , , and , , be the eigenvalues of the tridiagonal matrix … ►
31.5.2
►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities .
These solutions are the Heun polynomials.
…
2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
… ►Normalization
… ►Orthogonal Invariance
… ►Summation
►For , …3: 24.1 Special Notation
…
►
Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►
,
…
►
Euler Numbers and Polynomials
… ►The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …4: 18.3 Definitions
§18.3 Definitions
… ►With the property that is again a system of OP’s. See §18.9(iii).
Bessel polynomials
…5: 18.35 Pollaczek Polynomials
6: 18.30 Associated OP’s
…
►
§18.30(v) Associated Meixner–Pollaczek Polynomials
… ►They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10). … ►The type 3 Pollaczek polynomials are the associated type 2 Pollaczek polynomials, see §18.35. The relationship (18.35.8) implies the definition for the associated ultraspherical polynomials .7: 27.19 Methods of Computation: Factorization
…
►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored.
…
►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard algorithm, and the Elliptic Curve Method (ecm).
…
►Type II probabilistic algorithms for factoring rely on finding a pseudo-random pair of integers that satisfy .
…As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer.
…The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs.
…
8: 18.39 Applications in the Physical Sciences
…
►The recursion of (18.39.46) is that for the type 2 Pollaczek polynomials of (18.35.2), with , , and , and terminates for being a zero of the polynomial of order .
Thus the and the eigenvalues
…are determined by the zeros, of the Pollaczek polynomial
.
►
The Coulomb–Pollaczek Polynomials
►The polynomials , for both positive and negative , define the Coulomb–Pollaczek polynomials (CP OP’s in what follows), see Yamani and Reinhardt (1975, Appendix B, and §IV). …9: Bibliography P
…
►
A Kummer-type transformation for a hypergeometric function.
J. Comput. Appl. Math. 173 (2), pp. 379–382.
…
►
Several new product identities in relation to Rogers-Ramanujan type sums and mock theta functions.
Res. Math. Sci. 8 (1), pp. Paper No. 16, 58.
…
►
Sur une généralisation des polynomes de Legendre.
C. R. Acad. Sci. Paris 228, pp. 1363–1365.
►
Systèmes de polynomes biorthogonaux qui généralisent les polynomes ultrasphériques.
C. R. Acad. Sci. Paris 228, pp. 1998–2000.
…
►
Voronoi type congruences for Bernoulli numbers.
In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),
…