Computation of Some Functions of Two Variables Using Continued Fractions

V. E. Kornilov

(Presented by the Department of Higher Mathematics)

The article proves the possibility of representing certain hypergeometric functions of two variables as the sum of a suitable fraction and a remainder term. In this case the remainder term, in modulus, is less than the modulus of the difference of two consecutive approximants, in the order of increasing indices of the suitable fractions, in the complex domain \(\operatorname{Re} z \le 0\).

1. We write the functions of two variables ([1], pp. 219–220) in the following form:

\[ \left\{ \begin{aligned} F_1 &= \sum_{n=0}^{\infty} \left[ \frac{(\alpha)_n(\beta_1)_n}{(\gamma)_n n!} \,F(\alpha+n,\beta;\gamma+n;x) \right] z^n \\ &= \sum_{n=0}^{\infty} a_n z^n;\quad \alpha<\gamma,\ \beta_1<1,\ \beta<0,\ -1\le x\le 1. \end{aligned} \right. \tag{1} \]

\[ \left\{ \begin{aligned} (1-x)^3 F_2 &= \sum_{n=0}^{\infty} \left[ \frac{(\alpha)_n(\beta_1)_n}{(\gamma_1)_n n!} \,F\!\left(\gamma-\alpha-n,\beta;\gamma;\frac{x}{x-1}\right) \right] z^n \\ &= \sum_{n=0}^{\infty} b_n z^n;\quad \alpha<\gamma_1,\ \beta_1<1,\ \beta>0,\ -\infty<x\le \frac12 . \end{aligned} \right. \tag{2} \]

\[ \left\{ \begin{aligned} (1-x)^\beta F_3 &= \sum_{n=0}^{\infty} \left[ \frac{(\alpha_1)_n(\beta_1)_n}{(\gamma)_n n!} \,F\!\left(\gamma+n-\alpha,\beta;\gamma+n;\frac{x}{x-1}\right) \right] z^n \\ &= \sum_{n=0}^{\infty} c_n z^n;\quad \alpha_1<\gamma,\ \beta_1<1,\ \beta<0,\ -\infty<x\le \frac12 . \end{aligned} \right. \tag{3} \]

On the basis of equalities (1)—(3), the theorem ([2], p. 211), and also ([3], p. 226; [4], problem 1220), the determinants

\[ \left| \begin{array}{cccc} a_p & . & . & a_{p+m-1}\\ . & . & . & .\\ a_{p+m-1} & . & . & a_{p+2m-2} \end{array} \right|, \ .\ .\ .;\quad p,m=1,2,\ldots \]

are positive. Therefore the power series (1)|(3) are each represented as the sum of a suitable fraction and a remainder term. In this case the remain-

term, in modulus, is less than the modulus of the difference of two neighboring convergents in the domain \(\operatorname{Re} z \leqslant 0\) [5].

When establishing the ranges of variation of the parameters of the functions \(F\) in equalities (1)—(3), it is necessary to choose the parameter values so that, in the limiting case \(x=1\) or \(x \to \infty\), these functions decrease monotonically as the index \(n\) increases [5].

For example, for the function \(F\) of series (1) we have ([6], p. 374, (67)):

\[ F(\alpha+n,\beta;\gamma+n;1)= \frac{\Gamma(\gamma+n)\Gamma(\gamma-\alpha-\beta)} {\Gamma(\gamma-\alpha)\Gamma(\gamma+n-\beta)}. \]

The sequence

\[ \left\{ \frac{\Gamma(\gamma+n)} {\Gamma(\gamma+n-\beta)} \right\} \quad n=0,\ 1,\ 2,\ldots \]

decreases monotonically only in the case when the inequality \(\beta<0\) is satisfied. For formulas (2) or (3), these conditions are respectively \(\beta>0\) or \(\beta<0\).

REFERENCES

1. H. Bateman and A. Erdélyi. Higher Transcendental Functions (hypergeometric function, Legendre functions). Moscow, Fizmatgiz, 1965.

2. I. P. Natanson. Theory of Functions of a Real Variable. Moscow, Gostekhizdat, 1957.

3. [[unclear: beginning of reference]] limits, series, continued fractions). Moscow, Fizmatgiz, 1961.
   limits, series, continued fractions). Moscow, Fizmatgiz, 1961.

4. I. V. Proskuryakov. A Collection of Problems in Linear Algebra. Moscow, Gostekhizdat, 1957.

5. V. E. Kornilov. Application of continued fractions to the computation of generalized hypergeometric functions. Izv. TPI, vol. 205, 1972.

6. V. I. Smirnov. A Course of Higher Mathematics. Vol. 3, part two. Moscow, Gostekhizdat, 1953.