Let p > 3 be a prime, and let Rp be the set of rational numbers whose denominator is not divisible by p. Let {Pn(x)} be the Legendre polynomials. In this paper we mainly show that for m,n, t ∈ Rp with m 6≡ 0 (mod p), P[ p 6 ](t) ≡ − (3 p ) p−1 ∑ x=0 (x3 − 3x + 2t p ) (mod p)

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a para...

Features for a new application in pattern recognition are presented. The objects to be classiied are coccoliths (marine microfos-sils). Since the objects diier both in their outline and in their internal structure the features developed take into consideration both kinds of variability. The features concerning the internal structure are computed on the grey-value images produced by a scanning e...

We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuou...

We give a general expansion formula of functions in the Askey–Wilson polynomials and using the Askey–Wilson orthogonality we evaluate several integrals. Moreover we give a general expansion formula of functions in polynomials of Askey–Wilson type, which are not necessarily orthogonal. Limiting cases give expansions in little and big q-Jacobi type polynomials. We also give a new generating funct...

For t ≥ 0, let X(t) = (X0(t), . . . , Xp(t)) , where Xi(t) denotes the integral of the ith order Legendre polyonimal with respect to the same Brownian motion described by the corresponding standard deviation (0 ≤ i ≤ p). We obtain the exact tail behavior of P ( sup0≤t≤h |X(t)| > u ) as u → ∞, and the limit distribution of sup0≤t≤T |X(t)| as T → ∞. These processes naturally arise in the context ...

We rst generalize the Schur congruence for Legendre polynomials to sequences of polyno-mials that we call \d-Carlitz". This notion is more general than a similar notion introduced by Carlitz. Then, we study automaticity properties of double sequences generated by these sequences of polynomials, thus generalizing previous results on double sequences produced by one-dimensional linear cellular au...

We extend the definition of the classical Jacobi polynomials withindexes α,β > −1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the...

Abstract. Let p > 3 be a prime, and let m be an integer with p ∤ m. In the paper we solve some conjectures of Z.W. Sun concerning Pp−1 k=0 2k k 3 /mk (mod p2), Pp−1 k=0 2k k 4k 2k /mk (mod p) and Pp−1 k=0 2k k 2 4k 2k /mk (mod p2). In particular, we show that P p−1 2 k=0 2k k 3 ≡ 0 (mod p2) for p ≡ 3, 5, 6 (mod 7). Let {Pn(x)} be the Legendre polynomials. In the paper we also show that P[ p 4 ]...

Two sets of the Heun functions are introduced via integrals. Theorems about expanding functions with respect to these sets are proven. A number of integral and series representations as well as integral equations and asymptotic formulas are obtained for these functions. Some of the coefficients of the series are orthogonal (J-orthogonal) functions of discrete variables and may be interpreted as...