COMPUTATION OF THE GAMMA FUNCTION USING CONTINUED FRACTIONS

V. E. KORNILOV

(Presented by the Department of Higher Mathematics)

In the article a generalization is obtained of known formulas representing the functions \(\psi(1+z)\) and \(\ln \Gamma(1+z)\) by power series. The infinite power series occurring in these formulas are represented by continued fractions with positive link terms. If these power series are replaced by the sum of a suitable approximant and a remainder term, then the modulus of the remainder term is less than the modulus of the difference of two approximants whose indices differ by an odd integer. This estimate is valid in a certain domain of the complex variable \(z\).

1. If in the equality ([1], p. 60)

\[ \psi(1+z)=-\gamma+\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{z+n}\right),\quad \gamma=0.577\ldots \tag{1} \]

we use the expansion

\[ \frac{1}{\kappa}-\frac{1}{z+\kappa}=\frac{z}{\kappa^{2}}-\frac{z^{2}}{\kappa^{3}}+\frac{z^{3}}{\kappa^{4}}-\ldots,\quad |z|<\kappa, \]

then we obtain

\[ \psi(1+z)=-\gamma+\sum_{n=1}^{\kappa-1}\left(\frac{1}{n}-\frac{1}{z+n}\right)+ \]

\[ +\sum_{n=2}^{\infty}(-1)^{n}\zeta(n,\kappa)z^{n-1},\quad |z|<\kappa, \tag{2} \]

where

\[ \zeta(n,\kappa)=\sum_{m=\kappa}^{\infty}\frac{1}{m^{n}}. \tag{3} \]

Integrating equality (2), we find ([1], p. 30, (1))

\[ \ln \Gamma(1+z)=-\gamma z+\sum_{n=1}^{\kappa-1}\left[\frac{z}{n}-\ln\left(1+\frac{z}{n}\right)\right]+ \]

\[ +\sum_{n=2}^{\infty}(-1)^{n}\zeta(n,\kappa)\frac{z^{n}}{n},\quad |z|<\kappa. \tag{4} \]

After exponentiation we obtain

\[ \Gamma(1+z)=e^{-\gamma z}\prod_{n=1}^{k-1}\left(1+\frac{z}{n}\right)^{-1}\cdot e^{z/n}\times \]

\[ {}\times \exp\left[\sum_{n=2}^{\infty}(-1)^n\zeta(n,k)\frac{z^n}{n}\right],\quad k=1,2,\ldots \tag{5} \]

In equality (5) replace \(z\) by \(-z\) and divide it by the new equality; then apply the identity \(\pi z=\Gamma(1+z)\Gamma(1-z)\sin\pi z\), and after elementary transformations we finally obtain

\[ \Gamma(1+z)=e^{-\gamma z}\sqrt{\frac{\pi z}{\sin\pi z}}\prod\sqrt{\frac{n-z}{n+z}}\,e^{z/n}\times \]

\[ {}\times \exp\left[-\sum_{n=1}^{\infty}\zeta(2n+1,k)\frac{z^{2n+1}}{2n+1}\right],\quad |z|<k,\quad k=1,2,\ldots \tag{6} \]

As \(k\to\infty\) we have

\[ \Gamma(1+z)=e^{-\gamma z}\sqrt{\frac{\pi z}{\sin\pi z}}\prod_{n=1}^{\infty}\sqrt{\frac{n-z}{n+z}}\,e^{z/n},\quad |z|<\infty . \tag{7} \]

For \(z=1,2,\ldots\), in the right-hand side of equality (7) it is necessary to pass to the limit.

The power series in equality (5) is represented by a continued fraction with positive partial numerators \([2,3]\), and for it the remainder term is, in modulus, less than the modulus of the difference of two convergents whose indices differ by an odd number, in the region \(\operatorname{Re} z<0\) \([3]\). An analogous continued fraction can also be obtained for the power series (6) (\([4]\), problem 1220); here the corresponding estimate of the remainder term holds in the region

\[ \frac{\pi}{4}<|\arg z^2|<\frac{3}{4}\pi . \]

Representations of functions analogous to equalities (2), (4)—(6) can be obtained for hyperbolic and trigonometric functions, for logarithms of these functions, and for integrals of any of the listed functions and for integrals of the product of any of these functions by various powers of the variable of integration. The power series obtained under such a transformation are represented by continued fractions with positive partial numerators, and analogous estimates of the remainder terms in modulus are valid for them.

1. The generalized Riemann zeta-function (3) is represented by an integral of trigonometric and hyperbolic functions \([2]\); therefore it will be appropriate here to give a brief account of the question of applying continued fractions to the computation of trigonometric series.

When a trigonometric series is expanded in powers of \(e^{ix}\) or \(e^{-ix}\), it can be regarded as an ordinary power series. Such a series can almost always either be represented directly by a continued fraction with positive partial numerators, or be represented by several sums of infinite series, each of which is represented by a continued fraction with positive partial numerators. The exceptions are trigonometric series whose coefficients contain factors that are functions of the form \(\sin n\lambda\), \(\cos n\lambda\). However, such series also are transformed into a finite number of sums of infinite series possessing the property indicated above.

with the property that, if the trigonometric functions \(\sin n\lambda\), \(\cos n\lambda\) are replaced by exponential functions according to Euler’s formulas and transformations analogous to the transformations of the simplest trigonometric series are applied.

References

1. G. Beitmen and A. Erdélyi. Higher Transcendental Functions (Hypergeometric Function. Legendre Functions). Moscow, Fizmatgiz, 1965.

2. V. E. Kornilov. Application of continued fractions to the computation of Riemann’s zeta function. Izv. TPI, vol. 245.

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

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