# The algorithms of constructing the continued fractions for any rations of the hypergeometric Gaussian functions

2017;
: pp. 48-58

Received: June 15, 2017
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University

An algorithm for constructing recurrence relations of geometric Gaussian functions, in which the displacement of parameters is equal to $0$, $1$ or $-1$, is described. On the basis of such recurrence relations, the expansion for the ratio of Gaussian functions into continued fractions is developed. The obtained continued fractions are the development of the corresponding hypergeometric Gaussian functions in the case when the parameters of the function are integers.

1. Abramovvitz M., Stegun I. A. Handbook of mathematical functions with formulas, grapth and mathematical tables. NBS (1972).
2. Bateman H., Erdélyi A. Higher transcendental functions. Vol.1, Moscow, Nauka, 295 p. (1973), (in Russian).
3. Lebedev N. Special functions and their applications. Mosсow-Leningrad, Fizmatgiz, 630 p. (1963), (in Rusiian).
4. Luke Y. Special mathematical functions and their approximation. Moscow, Mir, 608 p. (1980), (in Rusiian).
5. Chuluunbaatar O. Mathematical models and logarithms for the analysis of processes of ionization of helium atoms and hydrogen molecules with variational functions. Bulletin of the TvGU. Series: Applied Mathematics. 47--64 (2008), (in Rusiian).
6. Exton H. Multiple hypergeometric functions and applications. New York, Sydney, Toronto, Chichester, Ellis Hoorwood, 376 p. (1976).
7. Verlan A., Sizikov V. Integral equation methods, algorithms, programs. Kyiv, Naukova Dumka, 544 p. (1986), (in Ukrainian).
8. Popov B., Tesler H. The calculation functions on the computer: Directory. Kyiv, Naukova Dumka, 600 p. (1984), (in Russian).
9. Manziy O., Hladun V., Pabirivsky V., Uhanska O. Algorithms for calculating the value of some hypergeometric Gaussian function in the complex plane. Physical and mathematical modeling and information technologies. Iss.19, 17–26 (2014), (in Ukrainian).
10. Cuyt A., Petersen V. B., Verdonk B., Waadeland H., Jones W. B. Handbook of Continued Fractions for Special Functions. Berlin, Springer, 431 p. (2008).
11. William B. J., Thron W. J.  Continued fractions. Analytic theory and applications. Vol.2. Moscow, Mir, 414 p. (1985), (in Russian).
12. Lorentzen L., Waadelamd H. Continued Fractions. Convergence Theory. Atlantis Press World Scientific, Amsterdam, Paris, 308 p. (2008).
Math. Model. Comput. Vol.4, No.1, pp.48-58 (2017)