The Basic Analogue of Kummer’s Theorem
نویسندگان
چکیده
which has since been known as Kummer's theorem. This appears to be the simplest relation involving a hypergeometric function with argument ( — 1). All the relations in the theory of hypergeometric series rF8 which have analogues in the theory of basic series are those in which the argument is ( + 1 ) . Apparently, there has been no successful at tempt to establish the basic analogue of any formula involving a function rF8( — l). Since Kummer's theorem is fundamental in the proofs of numerous relations between hypergeometric functions of argument ( — 1), it seemed desirable that an at tempt be made to prove the basic analogue of Kummer's theorem and to investigate the possibility of obtaining new relations in basic series with arguments corresponding to the argument ( — 1) in the classical case. In this paper, the basic analogue of Kummer's theorem is obtained
منابع مشابه
Kummer’s Special Case of Fermat’s Last Theorem∗
One particularly elegant example of an application of modern algebraic number theory to a classical problem about the integers is found in Kummer’s special case of Fermat’s Last Theorem. In this paper, we reduce Fermat’s Last Theorem to the question of whether or not there exist integer solutions to xp + yp = zp for p an odd prime. We then give a thorough exposition of Kummer’s proof that no su...
متن کاملA proof of Kummer’s theorem
Following suggestions of T. H. Koornwinder [3], we give a new proof of Kummer’s theorem involving Zeilberger’s algorithm, the WZ method and asymptotic estimates. In the first section, we recall a classical proof given by L. J. Slater [7]. The second section discusses the new proof, in the third section sketches of similar proofs for Bailey’s and Dixon’s theorems are given. The author is gratefu...
متن کاملNew Laplace transforms of Kummer's confluent hypergeometric functions
In this paper we aim to show how one can obtain so far unknown Laplace transforms of three rather general cases of Kummer’s confluent hypergeometric function 1F1(a; b; x) by employing generalizations of Gauss’s second summation theorem, Bailey’s summation theorem and Kummer’s summation theorem obtained earlier by Lavoie, Grondin and Rathie. The results established may be useful in theoretical p...
متن کاملThe Basic Theorem and its Consequences
Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a...
متن کاملKummer’s Criterion on Class Numbers of Cyclotomic Fields
Kummer’s criterion is that p divides the class number of Q(μp) if and only if it divides the numerator of some Bernoulli number Bk for k = 2, 4, . . . , p − 3. This talk will start with explaining how finite groups of Dirichlet characters are in bijection with finite Abelian extensions of Q, and why the class number of an Abelian CM field is “almost” computable. This computation involves the ge...
متن کامل