Appell Polynomials Research Articles

A Note on Generalized Appell Polynomials

A Note on Generalized Appell Polynomials

Expansions of $t$ Densities and Related Complete Integrals

A class of alternatives is here presented to Fisher's (1925) expansion of Student's $t$ density function. These expansions involve Appell's polynomials; and hence, recurrence schemes are available for the coefficients. Complete integrals of products of $t$ densities are of interest as Behrens-Fisher densities (viewed as Bayesian posterior distributions: Jeffreys, 1940; Patil, 1964) as moments of Bayesian posterior distributions (Anscombe, 1963; Tiao and Zellner, 1964). A symptotic expansion of complete integrals, obtained by term-by-term integration of these expansions, are favorably compared with those obtained from Fisher's expansion. Although expansions of complete integrals of products of multivariate $t$ densities can be developed from these expansions by the methods of Tiao and Zellner, the resulting coefficients are practically as complicated as the Tiao and Zellner coefficients; methods will be published soon (Dickey, 1965) for reducing the dimensionality of such integrals for quadrature. The paper concludes with a numerical study of the integral expansions.

Appell Polynomials of Large Order

Appell Polynomials of Large Order

Exponential Transforms and Appell Polynomials

Exponential Transforms and Appell Polynomials

A differential equation for Appell polynomials

A differential equation for Appell polynomials

Expansions in Generalized Appell Polynomials, and a Class of Related Linear Functional Equations

Expansions in Generalized Appell Polynomials, and a Class of Related Linear Functional Equations

Expansions in generalized Appell polynomials, and a class of related linear functional equations

where the Pn(x) are polynomials of bounded degree (< k), and the relation of this equation to expansions in generalized Appell polynomials. Equation (1) is transformed into an equivalent contour integral equation, using the Laplace transformation; and this integral equation is then shown to lead to a solution which is itself a contour integral, with a kernel which satisfies a linear differential equation of order k. It is also shown that the contour integral equation is equivalent to an expansion question in generalized Appell polynomials. In ?2 we derive some simple properties of these polynomials. In ?3 the equivalence between the above-mentioned differential and integral equations is shown, and the relation to the expansion problem developed. The resolving kernel for the general case (k) is introduced in ?4, and the particular cases k =0, k =1, are treated in ??5, 6. The method of the Laplace transformation is then extended, in ??7, 8 respectively, to partial differential equations of infinite order, constant coefficients, and to Laurent differential equations, constant coefficients.: 2. Generalized Appell polynomials. Let