Applications of Volterra's theory of composition to hypergeometric functions
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The main results of this thesis are the generalizations of previous known results given by Hadamard (6), and Thielman (12) on integral addition theorems of Bessel functions, namely Theorems 5.8 and 5.9. Also an integral addition theorem, Theorem 5.10, for Laguerre polynomials is obtained;The methods used are based on the isomorphism which exists between Volterra's theory of permutable functions and the algebra of polynomials and power series. From known algebraic relations between certain given functions, integral addition theorems are obtained for new functions which are the Volterra transforms (see Definition 3.1) of the given functions. In particular recursion formulas for Tchebycheff polynomials lead to the integral addition theorems mentioned above;Other applications of the theory of composition are given, some of these lead to the evaluation of certain integrals and series expansions for hypergeometric functions. Identities concerning Tchebycheff polynomials are derived on the basis of the commutative property (see Definition 4.2) of these polynomials. Also an expression for a series involving triple products of Tchebycheff polynomials is obtained directly from the generating function of these polynomials. Since the Volterra transform of zp Tn(1 - 2z) is 2F2(n,-n;½,p;y - x), p > 0 some properties of the set of polynomials 2F2(n,-n;½,p;t), (n = 0,1,2,…), are obtained from these identities and theorems on Tchebycheff polynomials.