[en] The definite integrals ∫_-1¹ P_l(x) P_λ^μ(x) dx are calculated by explicitly writing the Legendre and their associated polynomials in a suitable form. Selection rules derived from this approach are obtained. (Author)

[en] Bilateral generating functions are those involving products of different types of polynomials. It is showed that operational methods offer a powerful tool to derive these families of generating functions. Cases relevant to products of Hermite and Laguerre, Hermite and Legendre ... polynomials and propose further extensions of the method are studied in this report

[en] New nonlinear connection formulae of the q-orthogonal polynomials, such as continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and q-Gegenbauer polynomials, in terms of their respective classical analogues are obtained using a special realization of the q-exponential function as infinite multiplicative series of ordinary exponential function

[en] The asymptotic density of the zeros of Laguerre and Jacobi polynomials of diverging index and/or order is computed. (author)

[en] Using a realization of the q-exponential function as an infinite multiplicative series of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective classical analogues

[en] Laguerre, Hermite and Legendre polynomial bases were studied for high order time expansions of reactor kinetics solutions. A theorem showing an exponential majoring function for the solution of bounded reactivity transients introduce Laguerre, Hermite and Legendre polynomials for semi-infinite, infinite and finite time domains, respectively. The numerical solutions were obtained by means of the construction of an error estimator and its minimization using a conventional variational method. Some point reactor kinetics problems with exact solution were tested. The results showed a numerical monotone convergent behavior and accuracy, but problem-dependent efficiency caused by the extremely large expansion orders (more than 200 terms) needed in the studied bases for the cases with large reactivity insertions. (author)

[en] In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials

[en] This paper deals with an extension of the generating functions of the Hermite polynomials found of which these polynomials are particular solutions

[en] Formulae expressing explicitly the Jacobi coefficients of a general-order derivative (integral) of an infinitely differentiable function in terms of its original expansion coefficients, and formulae for the derivatives (integrals) of Jacobi polynomials in terms of Jacobi polynomials themselves are stated. A formula for the Jacobi coefficients of the moments of one single Jacobi polynomial of certain degree is proved. Another formula for the Jacobi coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given. A simple approach in order to construct and solve recursively for the connection coefficients between Jacobi-Jacobi polynomials is described. Explicit formulae for these coefficients between ultraspherical and Jacobi polynomials are deduced, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Two analytical formulae for the connection coefficients between Laguerre-Jacobi and Hermite-Jacobi are developed

[en] A simple method to generate leading coefficients for high-order sets of orthogonal polynomials, by derivation of recurrence expression for these coefficients, is developed. The method is applied to Legendre, Hermite, Chebyshev and Laguerre polynomials. The method may be used in calculations of high anisotropic neutron-scattering transfer cross-sections, where the angular distribution of the scattered neutrons is given in the ENDF/B files for most materials as coefficients of an expansion into Legendre polynomials. (author)