Application of Continued Fractions to the Computation of MacRobert’s E-Function

V. E. Kornilov

(Submitted by the Department of Higher Mathematics of the Tomsk Polytechnic Institute)

In the present article, MacRobert’s \(E\)-function (1) is represented in the form of a sum of convergents and a remainder term, for which an estimate is given in the domain of the real variable \(x>0\).

In the article the following abbreviated notation is used:
\[ y=x:(z\ldots t),\qquad (a)_n=a(a+1)\ldots(a+n-1),\qquad (a)_0=1. \]

For MacRobert’s \(E\)-function ([1], p. 200)
\[ E(\alpha_1,\ldots,\alpha_p,\;1:\rho_1,\ldots,\rho_q:x)=E[x]= \]
\[ =\sum_{m=0}^{\infty} \frac{\Gamma(\alpha_1+m)\ldots\Gamma(\alpha_p+m)} {\Gamma(\rho_1+m)\ldots\Gamma(\rho_q+m)} \left(-\frac{1}{x}\right)^m,\qquad x\gg 1,\quad p>q+1, \tag{1} \]
whose parameters satisfy the inequalities
\[ \alpha_1>\ldots>\alpha_p>1,\quad \rho_1>\ldots>\rho_q, \]
\[ \rho_1>\alpha_{p-q+1},\ldots,\quad \rho_q>\alpha_p, \tag{2} \]
the following theorem holds.

Theorem. The function (1) with parameters (2) can be represented in the form of the following \(p-1\)-fold sum of convergents:
\[ E[x]=\frac{\Gamma(\alpha_1)\ldots\Gamma(\alpha_p)} {\Gamma(\rho_1)\ldots\Gamma(\rho_q)} \times \]
\[ \times \sum_{m=1}^{n}\cdots\sum_{\nu=1}^{n} b_{1m}\ldots b_{p-1,\nu}\, \frac{P_{2n}(x_{p-1})}{Q_{2n}(x_{p-1})} +R_{pn}(x). \tag{3} \]

The coefficients \(a_{\pi\mu}, b_{\pi\mu}\) \((\pi=1,\ldots,p-1,\ \mu=1,\ldots,n)\) in equality (3) all have positive values ([2], pp. 5, 24) and, in view of ([3], p. 28, (7)), ([4], p. 23, (5)), are determined according to the equalities
\[ \left\{ \begin{aligned} &\prod_{\mu=1}^{n}\left(1+\frac{x}{a_{\pi\mu}}\right) =\sum_{\kappa=0}^{n} C_n^\kappa \frac{x^\kappa}{(a_\pi)_\kappa} =q_{2n}(x),\\ &\qquad a_{\pi1}>\ldots>a_{\pi n}>0,\quad \pi=1,\ldots,p-q,\\[6pt] &\prod_{\mu=0}^{n}\left(1+\frac{x}{a_{\pi\mu}}\right) =\sum_{\kappa=0}^{n} C_n^\kappa \frac{(\rho_{\pi-p+q}+n-1)_\kappa}{(a_\pi)_\kappa} x^\kappa =Q_{2n}(x),\\ &\qquad 1>a_{\pi1}>\ldots>a_{\pi n}>0,\quad \pi=p-q+1,\ldots,p-1. \end{aligned} \right. \tag{4} \]

\[ \left\{ \begin{aligned} b_{\pi\mu}&=\frac{-p_{2n}(-a_{\pi\mu})} {a_{\pi\mu}\dfrac{d}{dx}\,[q_{2n}(-a_{\pi\mu})]}, \qquad \pi=1,\ldots,p-q;\\[6pt] p_{2n}(x)&=\sum_{\kappa=0}^{n-1}x^{\kappa+1} \sum_{m=0}^{\,n-\kappa-1} C_n^m \frac{(-1)^{\,n-\kappa-m-1}} {(a_\pi+n-\kappa-m-1)_{\kappa+1}},\\[6pt] b_{\pi\mu}&=\frac{-p_{2n}(-a_{\pi\mu})} {a_{\pi\mu}\dfrac{d}{dx}\,[q_{2n}(-a_{\pi\mu})]}, \qquad \pi=p-q+1,\ldots,p-1; \quad P_{2n}(x)=\\[4pt] &=\sum_{\kappa=0}^{n-1}x^{\kappa-1} \sum_{m=0}^{\,n-\kappa-1} C_n^{\kappa+m+1}(-1)^m \frac{(\rho_{\pi-p+q}+n-1)_{\kappa+m+1}} {(\alpha_\pi+m)_{\kappa+1}(\rho_{\pi-p+q})_m}. \end{aligned} \right. \tag{5} \]

The remainder term \(R_{pn}(x)\) in equality (3) is less than the sum of \(0,1,\ldots,p-1\)-fold sums:

\[ \begin{aligned} R_{pn}(x)<\frac{\Gamma(\alpha_1)\ldots\Gamma(\alpha_p)} {\Gamma(\rho_1)\ldots\Gamma(\rho_q)} \Biggl[ &\frac{n!}{(\alpha_1+1)_n q_{2n}(x_0)q_{2n+1}(x_0)} +\\ &+\sum_{m=1}^{n}\frac{b_{1m}n!} {(\alpha_2+1)_n q_{2n}(x_1)q_{2n+1}(x_1)} +\cdots+\\ &+\sum_{m=1}^{n}\cdots\sum_{j=1}^{n} \frac{b_{1m}\cdots b_{p-q-1,j}n!} {(\alpha_{p-q}+1)_n q_{2n}(x_{p-q-1})q_{2n+1}(x_{p-q+1})} +\cdots+\\ &+\sum_{m=1}^{n}\cdots\sum_{\nu=1}^{n} \frac{b_{1m}\cdots b_{p-1,\nu}n!(\rho_q-\alpha_p)_n} {(\rho_q)_n(\alpha_p+1)_n Q_{2n}(x_{p-1})Q_{2n+1}(x_{p-1})} \Biggr]. \end{aligned} \tag{6} \]

In inequality (6) the polynomials \(q_{2n+1}(x)\) and \(Q_{2n+1}(x)\), and the coefficients \(x_0,\ldots,x_{p-1}\), are as follows:

\[ q_{2n+1}(x)=\sum_{\kappa=0}^{n} C_n^\kappa \frac{x^\kappa}{(\alpha_\pi+1)_\kappa}, \qquad \pi=1,\ldots,p-q. \tag{7} \]

\[ Q_{2n+1}(x)=\sum_{\kappa=0}^{n} C_n^\kappa \frac{(\rho_{\pi-p+q}+n)_\kappa}{(\alpha_\pi+1)_\kappa}x^\kappa, \qquad \pi=p-q+1,\ldots,p-1. \]

\[ x_0=\frac{\rho_1\ldots\rho_q}{\alpha_2\ldots\alpha_p}\,x,\qquad x_1=\frac{\rho_1\ldots\rho_q}{\alpha_3\ldots\alpha_p a_{1m}}\,x,\ldots, \tag{8} \]

\[ x_{p-q-1}=\frac{\rho_1\ldots\rho_q x} {\alpha_{p-q+1}\ldots\alpha_p a_{1m}\ldots a_{p-q-1,j}}, \ldots,\qquad x_{p-1}=\frac{x}{a_{1m}\ldots a_{p-1,\nu}}. \]

Proof. On the basis of ([1], p. 201, (3)) and p. 72, (10), (12), we represent the function (1) by a multiple integral:

\[ \begin{aligned} E[x]=I_p &=\frac{1} {\Gamma(\rho_1-\alpha_{p-q+1})\ldots\Gamma(\rho_q-\alpha_p)} \int_0^1 t^{\alpha_p-1}\times\\ &\quad\times(1-t)^{\rho_q-\alpha_p-1}\,dt\ldots \int_0^1 s^{\alpha_{p-q+1}-1}(1-s)^{\rho_1-\alpha_{p-q+1}-1}\,dx\times\\ &\quad\times\int_0^\infty r^{\alpha_{p-q}-1}e^{-r}\,dr\ldots \int_0^\infty v^{\alpha_1-1} \left(1+\frac{v\ldots t}{x}\right)^{-1}e^{-v}\,dv. \end{aligned} \tag{9} \]

The last integral in equality (9), in view of ([5], p. 144) and ([3], p. 28), shall be represented as the sum of a convergent fraction and a remainder term:

\[ E|x|=I_{p-1}\int_0^\infty v^{\alpha_1-1}\left(1+\frac{v\ldots t}{x}\right)^{-1}e^{-v}\,dv<I_{p-1}\Gamma(\alpha_1)\times \]

\[ \times\left[\frac{p_{2n}(y_1)}{q_{2n}(y_1)} +\frac{n!}{(\alpha_1+1)_n q_{2n}(y_1)q_{2n+1}(y_1)}\right], \quad \alpha_1>1,\quad y_1=\frac{y}{w}. \tag{10} \]

The second term of inequality (10) is evaluated to the end and is less than the first term of inequality (6)

\[ \frac{I_{p-1}\Gamma(\alpha_1)n!}{(\alpha_1+1)_n q_{2n}(y_1)q_{2n+1}(y_1)} < \frac{\Gamma(\alpha_1)\ldots\Gamma(\alpha_p)} {\Gamma(\rho_1)\ldots\Gamma(\rho_q)} \times \]

\[ \times \frac{n!}{(\alpha_1+1)_n q_{2n}(x_0)q_{2n+1}(x_0)}, \quad y_1=\frac{y}{w}. \tag{11} \]

To prove inequality (11), in view of equalities (15), we transform its left-hand side

\[ \frac{I_{p-1}n!}{(\alpha_1+1)_n q_{2n}(y_1)q_{2n+1}(y_1)} = \frac{I_{p-2}\Gamma(\alpha_2)n!} {(\alpha_1+1)_n q_{2n}(y_2)q_{2n+1}(y_2)} + \]

\[ +I_{p-2}\Gamma(\alpha_2) \left\{ \sum_{\lambda=1}^{n}\frac{\alpha_1}{\gamma_\lambda}\beta_\lambda \left[ \lim_{\kappa\to\infty}\frac{p_{2\kappa}(y:\gamma_\lambda)} {q_{2\kappa}(y:\gamma_\lambda)} -\frac{1}{1+\gamma_\lambda:y_2} \right] -\right. \tag{12} \]

\[ \left. -\sum_{\lambda=1}^{n}\zeta_\lambda \left[ \lim_{\kappa\to\infty}\frac{p_{2\kappa}(y:\delta_\lambda)} {q_{2\kappa}(y:\delta_\lambda)} -\frac{1}{1+\delta_\lambda:y_2} \right]\right\} \le \]

\[ \le \frac{I_{p-2}\Gamma(\alpha_2)n!} {(\alpha_1+1)_n q_{2n}(y_2)q_{2n+1}(y_2)}; \quad y_2=\frac{y}{\alpha_2}. \]

To prove inequality (12), let us obtain several formulas.

1). Arrange the roots of the convergents in increasing order
\(\gamma_1<\ldots<\gamma_n,\ \delta_1<\ldots<\delta_n\); then, in view of ([2], p. 15) and (15), \(\gamma_\lambda>\delta_\lambda,\ \lambda=1,\ldots,n\)

\[ \frac{p_{2\kappa}(y:\gamma_\lambda)} {q_{2\kappa}(y:\gamma_\lambda)} \le \frac{p_{2\kappa}(y:\delta_\lambda)} {q_{2\kappa}(y:\delta_\lambda)}, \quad \kappa=1,2,\ldots,\quad 0\le y<\infty. \tag{13} \]

2). The sequences of even convergents are increasing sequences ([5], p. 12)

\[ \frac{p_{2\kappa}(y)}{q_{2\kappa}(y)} \le \frac{p_{2\kappa+2}(y)}{q_{2\kappa+2}(y)}, \quad \kappa=1,2,\ldots, \tag{14} \]

3). The second term of the right-hand side of inequality (10), in view of (4), (5), (7) and ([6], p. 374, (67)), is transformed as follows:

\[ \frac{n!}{(\alpha_1+1)_n q_{2n}(y_1)q_{2n+1}(y_1)} = \frac{p_{2n+1}(y_1)}{q_{2n+1}(y_1)} - \frac{p_{2n}(y)}{q_{2n}(y)} = \]

\[ = 1-\frac{\alpha_1}{y_1} \sum_{\lambda=1}^{n} \frac{\beta_\lambda}{1+\gamma_\lambda:y_1} - \sum_{\lambda=1}^{n} \frac{\zeta_\lambda}{1+\delta_\lambda:y_1} = \tag{15} \]

\[ = \sum_{\lambda=1}^{n} \frac{\alpha_1\beta_\lambda:\gamma_\lambda}{1+\gamma_\lambda:y_1} - \sum_{\lambda=1}^{n} \frac{\zeta_\lambda}{1+\delta_\lambda:y_1} + \frac{n!}{(\alpha_1+1)_n}. \]

4) In view of inequalities (13) and (14), we obtain several more inequalities:

\[ \frac{y}{y+\gamma_\lambda \alpha_2} \leq \frac{y}{y+\gamma_\lambda(\alpha_2-\varepsilon)} \leq \frac{p_{2k+2}(y;\gamma_\lambda)}{q_{2k+2}(y;\gamma_\lambda)}, \quad k=1,2,\ldots \tag{16} \]

\[ \frac{y}{y+\delta_\lambda \alpha_2} \leq \frac{y}{y+\delta_\lambda(\alpha_2-\varepsilon)} \leq \frac{p_{2k+2}(y;\delta_\lambda)}{q_{2k+2}(y;\delta_\lambda)}, \quad \varepsilon>0. \tag{17} \]

Further, in view of inequalities (13), (14), (16), and (17), the sum enclosed in braces on the left-hand side of inequality (12) satisfies the inequality

\[ \begin{aligned} &\sum_{\lambda=1}^{n}\frac{\alpha_1}{\gamma_\lambda}\beta_\lambda \left[ \lim_{k\to\infty} \frac{p_{2k}(y;\gamma_\lambda)}{q_{2k}(y;\gamma_\lambda)} - \frac{y}{y+\gamma_\lambda \alpha_2} \right] - \sum_{\lambda=1}^{n}\zeta_\lambda \left[ \lim_{k\to\infty} \frac{p_{2k}(y;\delta_\lambda)}{q_{2k}(y;\delta_\lambda)} - \frac{y}{y+\delta_\lambda \alpha_2} \right] \leq \\ &\leq \sum_{\lambda=1}^{n}\frac{\alpha_1}{\gamma_\lambda}\beta_\lambda \left[ \frac{y}{y+\gamma_\lambda(\alpha_2-\varepsilon)} - \frac{y}{y+\gamma_\lambda \alpha_2} \right] - \sum_{\lambda=1}^{n}\zeta_\lambda \left[ \frac{y}{y+\delta_\lambda(\alpha_2-\varepsilon)} - \frac{y}{y+\delta_\lambda \alpha_2} \right] =F(y). \end{aligned} \tag{18} \]

The right-hand side of inequality (18), in view of (4), (7), and (15), has a nonpositive value

\[ F(y)= \frac{n!}{(\alpha_1+1)_n\,q_{2n}\!\left(\dfrac{y}{\alpha_2-\varepsilon}\right) q_{2n+1}\!\left(\dfrac{y}{\alpha_2-\varepsilon}\right)} - \frac{n!}{(\alpha_1+1)_n\,q_{2n}\!\left(\dfrac{y}{\alpha_2}\right) q_{2n+1}\!\left(\dfrac{y}{\alpha_2}\right)} \leq 0, \tag{19} \]

therefore, in view of inequalities (18) and (19), inequality (12) is also valid.

Transforming, analogously to what was set forth above, the \((p-2)\)-fold integral \(I_{p-2}\) of the right-hand side of inequality (12), and also in view of ([7], p. 312, (9)), ([4], p. 23, (5)) and (9), we obtain inequality (11).

We represent the first term of the right-hand side of inequality (10) in the form of a sum of elementary fractions. The integral of each elementary fraction, in turn, will again be an integral of the same form as the last integral on the right-hand side of equality (9), etc.

Successive integration of the elementary fractions once and of the remainder terms to the end, and the representation of the integrals of the elementary fractions by suitable fractions and remainder terms, and then the next integration of the elementary fractions, etc., will lead us to formula (3) and to the estimate of the remainder term according to formula (6), as was required to be proved.

REFERENCES

1. H. Bateman and A. Erdélyi. Higher Transcendental Functions (hypergeometric function, Legendre functions). Moscow, Fizmatgiz, 1965.
2. T. I. Stieltjes. Studies on Continued Fractions. Moscow, ONTI, 1936.
3. V. E. Kornilov. Application of continued fractions to the computation of certain types of integrals. “Izv. TPI,” vol. 131, pp. 26–30, Tomsk, 1965.

1. V. E. Kornilov. Application of continued fractions to the computation of integrals of binomial differentials. “Izv. TPI,” vol. 131, pp. 21–25, Tomsk, 1965.

2. A. N. Khovanskii. Application of continued fractions and their generalizations to problems of approximate analysis. Moscow, GITTL, 1956.

3. V. I. Smirnov. A Course of Higher Mathematics. Vol. 3, Part 2, Moscow, GITTL, 1953.

4. N. N. Lebedev. Special Functions and Their Applications. Moscow, Gostekhizdat, 1953.