COMPUTATION OF MULTIPLE INTEGRALS OF RIEMANN FUNCTIONS USING CONTINUED FRACTIONS

V. E. KORNILOV

(Presented by the Department of Higher Mathematics)

The article studies the question of representing a power series, into which a multiple integral is expanded, by a continued fraction with positive elements of the links. The remainder term is, in absolute value, less than the absolute value of the difference of two suitable fractions in the interval \(-1 \le z < 0\).

1. The Riemann zeta function ([1], p. 47)

\[ \zeta(s)=\frac{1}{1-2^{-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^s}, \quad s>0 \tag{1} \]

is a special case of the function ([1], pp. 42, 43)

\[ \Phi(z,s,v)=\sum_{n=0}^{\infty}(v+n)^{-s}z^n,\quad -1\le z<1,\ v>0. \tag{2} \]

Since

\[ \Gamma(s)z^n=(v+n)^s\int_{0}^{\infty}e^{-vt}t^{s-1}(ze^{-t})^n\,dt,\quad s>0, \tag{3} \]

then from formula ([1], p. 15 (5))

\[ \Gamma(s)=(v+n)^s\int_{0}^{\infty}e^{-(v+n)t}t^{s-1}\,dt,\quad s>0 \tag{4} \]

one obtains the integral formula ([1], p. 43 (3))

\[ \Gamma(s)\Phi(z,s,v)=\int_{0}^{\infty}t^{s-1}e^{(1-v)t}(e^t-z)^{-1}\,dt,\quad s>0. \tag{5} \]

On the basis of equality (4) we have

\[ [\ln(v+n)]^{-s}\Gamma(s)=\int_{0}^{\infty}e^{-t\ln(v+n)}t^{s-1}\,dt,\quad v>1, \tag{6} \]

since \(e^{-t\ln(v+n)}=(v+n)^{-t}\), then

\[ \int_{0}^{\infty}\frac{t^{s-1}\,dt}{(v+n)^t} = \frac{\Gamma(s)}{\ln^s(v+n)},\quad v>1,\ s>0. \tag{7} \]

Taking equality (7) into account, we compute the integral of both sides of equality (2)

\[ I_1=\int_0^\infty\left[\sum_{n=0}^\infty \frac{z^n\,dt}{(v+n)^t}\right] =\sum_{n=0}^\infty \frac{z^n}{\ln(v+n)},\quad v>1; \]

\[ \left\{ \begin{aligned} I_1&=\int_0^\infty \frac{ds}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}e^{(1-v)t}}{e^t-z}\,dt =\sum_{n=0}^\infty \frac{z^n}{\ln(v+n)},\\ v&>1. \end{aligned} \right. \tag{8} \]

The integral \(I_1\) can be generalized

\[ \left\{ \begin{aligned} I_\nu&=\int_0^\infty \frac{z^{\nu-1}\,ds}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}e^{(1-v)t}}{e^t-z}\,dt =\sum_{n=0}^\infty \frac{z^n}{\ln^\nu(v+n)},\\ &v>1,\quad \nu>0. \end{aligned} \right. \tag{9} \]

2. By virtue of elementary integrals

\[ \int_0^\infty \frac{d\nu}{\ln^\nu(v+n)} =\frac{1}{\ln[\ln(v+n)]},\quad v>e; \]

\[ \int_0^\infty \frac{d\nu}{\ln^{\nu_1\nu}(v+n)} =\frac{1}{\ln[\ln^{\nu_1}(v+n)]},\quad v>e \]

and of integral (9), it is not difficult to represent in the form of a power series the following multiple integral

\[ \left\{ \begin{aligned} I_1&=\int_0^\infty u_2^{\nu-1}\,du_2 \int_0^\infty \frac{u_1^{\nu_1u_2-1}\,du_1}{\Gamma(u_1)} \int_0^\infty \frac{u^{u_1-1}e^{(1-v)u}}{e^u-z}\,du =\\ &=\sum_{n=0}^\infty \frac{z^n}{\ln^{\nu_1}\!\left[\ln^\nu(v+n)\right]} =\sum_{n=0}^\infty c_n z^n,\\ &\nu,\nu_1>0,\quad v>e. \end{aligned} \right. \tag{10} \]

The Gram determinant

\[ \Delta_k= \begin{vmatrix} c_0 & \cdots & c_k\\ \cdot & \cdot & \cdot\\ c_k & \cdots & c_{2k} \end{vmatrix} >0 \]

on the grounds that \(c_n\) are power moments

\[ c_n=\int_0^\infty u_2^{\nu-1}\,du_2 \int_0^\infty \frac{u_1^{\nu_1u_2-1}\,du_1}{\Gamma(u_1)} \int_0^\infty u^{u_1-1}e^{-(v+n)u}\,du \]

for the system \(1,e^{-u},\ldots,e^{-nu},\ldots\).

By virtue of \(\Delta_k>0\), with respect to series (10) the corresponding continued fraction ([2], p. 229) has positive link terms.

([3], pp. 23–30). For the integral (10), represented by the sum of a continued fraction and the remainder term \(R_{2k}\), we have

\[ I_1=\frac{P_{2k}}{Q_{2k}}+R_{2k},\qquad R_m \equiv R_m(z); \]

\[ |R_{2k}|< \left| \frac{P_{2k+1}}{Q_{2k+1}}-\frac{P_{2k}}{Q_{2k}} \right|,\qquad -1\le z<0 \]
([4], p. 63).

References

1. Beitmen and A. Erdélyi. Higher Transcendental Functions. Moscow, Fizmatgiz, 1965.
2. V. L. Danilov, A. N. Ivanova et al. Mathematical Analysis. Moscow, Fizmatgiz, 1965.
3. T. I. Stieltjes. Investigations on Continued Fractions. Moscow, ONTI, 1936.
4. B. P. Demidovich and I. A. Maron. Foundations of Computational Mathematics. Moscow, Fizmatgiz, 1963.