On the Values of Hypergeometric Functions with Different Irrational Parameters at Small Points

Nauka i Obrazovanie

P. L. Ivankov¹ 

[1] Bauman Moscow State Technical University;

关键词: estimates of linear forms;    simultaneous approximations;    generalized hypergeometric functions;    irrational parameters;   

DOI  :  10.7463/0914.0726862

来源: DOAJ

【 摘 要 】

In this paper we consider a problem concerning the linear independence of the values of generalized hypergeometric functions with different irrational parameters. To solve this problem it is impossible to apply directly Siegel's method known in the theory of transcendental numbers, since abovementioned functions do not belong to a special class of entire functions Siegel has introduced. The method of Siegel uses a Dirichlet principle to construct linear approximating form. Such a construction cannot be realized if the coefficients of Taylor series of the functions under consideration have \bad" denominators. We have exactly this circumstance in the case of hypergeometric functions with irrational parameters. There exists a modification of the method of Siegel, which allows us to apply the Dirichlet principle in this case also, but it is still unknown if this modification can be used in a situation where the varied parameters are irrational.

Usually in the case of irrational parameters one applies effective methods for constructing linear approximating form. If we apply such a construction in the case of different irrational parameters we shall have to confine ourselves by only two parameters and introduce the additional requirement: the difference of these parameters must be a rational number. The practice of considering similar problems for hypergeometric functions with irrational parameters shows that the use of simultaneous approximations leads as a rule to better results. In case of problems with different irrational parameters it is possible, for example, to waive the condition of rationality of the varied parameters difference.

In this paper, the effective construction of simultaneous approximations differs from previously proposed, which made it possible owing to some additional considerations of a technical nature to improve the arithmetic part of the method (which is, mainly, to attain an acceptable estimate of the least common denominator of the coefficients of the approximating polynomials). As a result the above mentioned unnecessary restriction on the varied parameters was dropped, but we have to restrict ourselves to the values of the functions only at the point with a small absolute value.

Remaining within the methods used in this paper one can obtain some other results which are not discussed in the article, however, increasing the number of varied parameters (if only to three) or rejecting the restriction on the values of the functions will require, probably, some new ideas.

【 授权许可】

Unknown