Quadruple Series Equations involving Heal Polynomials

Dual series equations involving generalized Laguerre polynomials

where α + β + 1 > β > 1 −m, σ + 1 > α + β > 0, m is a positive integer, and 0 < h < ∞, 0 ≤ b <∞, and h and b are finite constants. L n [(x + b)h] is a Laguerre polynomial, An are unknown coefficients, and f (x) and g(x) are prescribed functions. Srivastava [5, 6] has solved the following dual series equations: ∞ ∑ n=0 AnL (α) n (x) Γ(α+n+ 1) = f (x), 0 < x < a, (1.3) ∞ ∑ n=0 AnL (σ) n (x) Γ(α+n...

Existence of Solutions for Equations Involving Iterated Functional Series

This section collects the standard terminology and results used in the sequel (see [5]). Let I = [a,b] be an interval of real numbers. C1(I ,I), the set of all continuously differentiable functions from I into I , is a closed subset of the Banach Space C1(I ,R) of all continuously differentiable functions from I into R with the norm ‖ · ‖c1 defined by ‖φ‖c1 = ‖φ‖c0 +‖φ′‖c0 , φ∈ C1(I ,R) where ‖...

Differentiable Solutions of Equations Involving Iterated Functional Series

Let f be a self-mapping on a topological space X and f denote the mth iterate of f , that is, f f ◦ fm−1, f0 id, m 1, 2, . . . . Let C X,X be the set of all continuous self-mappings on X. Equations having iteration as their main operation, that is, including iterates of the unknown mapping, are called iterative equations. It is one of the most interesting classes of functional equations 1–4 , b...

Congruences involving Bernoulli polynomials

Let {Bn(x)} be the Bernoulli polynomials. In the paper we establish some congruences for Bj(x) (mod p n), where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for h(−sp) (mod p) (s = 3, 5, 8, 12) and the sum P k≡r (mod m) p k , where h(d) is the class number of the quadratic field Q(d) of discriminant d...

Fractional differential equations with Legendre polynomials