ESTIMATE OF THE REMAINDER TERM OF THE MACROBERT E-FUNCTION

V. E. KORNILOV

(Presented by the Department of Higher Mathematics)

This article studies the question of estimating, in modulus, the MacRobert E-function [1] for a complex variable in the domain \(\operatorname{Re} z \ge 0\).

1. Let the parameters of the MacRobert E-function ([1], p. 200)

\[ E=(\alpha_1,\ldots,\alpha_p,1:\rho_1,\ldots,\rho_q:z)=E[z]= \sum_{m=0}^{\infty} \frac{\Gamma(\alpha_1+m)\cdots\Gamma(\alpha_p+m)} {\Gamma(\rho_1+m)\cdots\Gamma(\rho_q+m)} \left(-\frac{1}{z}\right)^m,\quad p\ge q+1 \tag{1} \]

satisfy the following inequalities:

\[ \alpha_1>\cdots>\alpha_p>1,\quad \rho_1>\cdots>\rho_q; \]

\[ \alpha_{p-q+1}<\rho_1<2\alpha_{p-q+1},\ldots,\alpha_p<\rho_q<2\alpha_p; \tag{2} \]

then for the function (1) the following theorem holds.

Theorem. The modulus of the MacRobert E-function (1), whose parameters satisfy the inequalities (2), for \(\operatorname{Re} z>0\) is less than the following sum:

\[ \begin{aligned} |E[z]|< \frac{\Gamma(\alpha_1)\cdots\Gamma(\alpha_p)} {\Gamma(\rho_1)\cdots\Gamma(\rho_q)} \Bigg[ &1:\left|1+\frac{\alpha_1\cdots\alpha_p}{\rho_1\cdots\rho_q z}\right| \\ &+\sum_{m=1}^{p-q} \frac{|z|(\rho_1-1)\cdots(\rho_q-1)} {\alpha_1\cdots\alpha_m(\alpha_m-1)\cdots(\alpha_p-1)} \\ &+\sum_{m=1}^{q} \frac{ (\rho_m-\alpha_{p-q+m})\rho_1\cdots\rho_m(\rho_{m+1}-1)\cdots(\rho_q-1) \left|\dfrac{\rho_1\cdots\rho_{m-1}z}{\alpha_1\cdots\alpha_{p-q+m-1}}+\dfrac{1}{2}\right|^{-1} } { \dfrac{3}{2|z|}\alpha_1\cdots\alpha_{p-q+m} (\alpha_{p-q+m+1}-1)\cdots(\alpha_p-1) \left|\dfrac{\rho_1\cdots\rho_m z}{\alpha_1\cdots\alpha_{p-q+m}}+1\right|^2 } \Bigg]. \end{aligned} \tag{3} \]

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

\[ E[z]=I_p= \frac{1}{\Gamma(\rho_1-\alpha_{p-q+1})\cdots\Gamma(\rho_q-\alpha_p)} \int_{0}^{1} t^{\rho_1-1}\times \]

\[ \times (1-t)^{\rho q-\alpha p-1}\,dt\ldots \int_0^1 s^{\alpha p-q+1-1}(1-s)^{\rho-\alpha p-q+1-1}\,ds \times \]

\[ \times \int_0^\infty r^{\alpha p-q-1}e^{-r}\,dr\ldots \int_0^\infty \vartheta^{q-1}\left(1+\frac{\vartheta\ldots t}{z}\right)^{-1}e^{-\vartheta}\,d\vartheta . \tag{4} \]

The last integral in equality (4), in view of ([2], p. 144) and ([3], p. 28), we represent in the form of the sum of a suitable fraction and a functional series:

\[ E[z]=I_{p-1}\Gamma(\alpha_1)\left[\left(1+\frac{\alpha_1}{z_1}\right)^{-1} +\sum_{m=1}^{\infty}\frac{m!z_1}{(\alpha_1)_{m+1}Q_{2m}(z_1)Q_{2m+2}(z_1)}\right]; \tag{5} \]

\[ z_1=\frac{z}{\alpha_1 w\ldots t},\qquad (\alpha_m)=\frac{\Gamma(\alpha+m)}{\Gamma(\alpha)} . \]

The modulus of the second term of the right-hand side of equality (5) is computed to the end and is equal to

\[ \frac{|z|(\rho_1-1)\ldots(\rho_q-1)} {\alpha_1(\alpha_1-1)\ldots(\alpha_p-1)} \]

in view of the fact that for \(\operatorname{Re} z>0\) in the article ([3], p. 28) it was obtained that

\[ \sum_{m=1}^{\infty} \frac{m!}{(\alpha_1)_{m+1}|Q_{2m}(z)Q_{2m+2}(z)|} < \sum_{m=0}^{\infty}\frac{(2)_m}{(\alpha_1)_{m+2}} = \frac{1}{(\alpha_1-1)_2}. \tag{6} \]

The modulus of the first term of the right-hand side of equality (5), if the factor \(\Gamma(\alpha_1)\) is discarded, is equal to the modulus of the following \(E\)-function:

\[ |I_{p-1}|=\left|E\left(\alpha_2,\ldots,\alpha_p;\,1:\rho_1,\ldots,\rho_q;\frac{z}{\alpha_1}\right)\right|. \tag{7} \]

The last integral of the form (5) of function (7) is again expanded into a continued fraction and represented by two terms according to equality (5), etc. After a \(p-q\)-fold replacement of integrals of the form (5) by two terms according to equality (5), and after complete integration of the second terms, we obtain

\[ J_{q-1}\int_0^1 s^{\alpha_{p-q+1}-1}(1-s)^{\rho_1-\alpha_{p-q+1}-1} \left(1-\frac{\alpha_1\ldots\alpha_{p-q}s\ldots t}{z}\right)^{-1}\,ds = \]

\[ = J_{q-1} \frac{\Gamma(\alpha_{p-q+1})\Gamma(\rho_1-\alpha_{p-q+1})}{\Gamma(\rho_1)} \left[ \left(1-\frac{\alpha_1\ldots\alpha_{p-q+1}u\ldots t}{\rho_1z}\right)^{-1} +\right. \]

\[ \left. +\frac{z}{\alpha_1\ldots\alpha_{p-q}u\ldots t} \sum_{m=1}^{\infty} \frac{(\rho_1-\alpha_{p-q+1})_m\,m!(\alpha_{p-q+1})_m} {(\rho_1)_{2m}(\rho_1+m-1)_m\,(Q_{2m}Q_{2m+2}} \right]. \tag{8} \]

Since the last integral of the left-hand side of equality (8), in view of ([4], p. 312 (3)) and ([5], p. 21), is represented by a suitable fraction and a remainder in the form of a functional series.

The modulus of the second term of equality (8) is computed to the end and, in view of inequalities (2), equality (4), ([5], p. 24), and inequalities (12), is equal to the following expression:

\[ r_1= \frac{ 2|z|(\rho_1-\alpha_{p-q+1})\rho_1(\rho_2-1)\ldots(\rho_q-1) \left|\dfrac{z}{\alpha_1\ldots\alpha_{p-q}}+\dfrac{1}{2}\right|^{-1} }{ 3\alpha_1\ldots\alpha_{p-q+1}(\alpha_{p-q+2}-1)\ldots(\alpha_p-1) \left|\dfrac{\rho_1z}{\alpha_1\ldots\alpha_{p-q+1}}+1\right|^2 }. \tag{9} \]

The modulus of the first term of equality (8), if the factor
\(\dfrac{\Gamma(a_1)\cdots\Gamma(a_{p-q+1})}{\Gamma(\rho_1)}\) is omitted, is equal to the modulus of the following \(E\)-function

\[ |I_{q-1}|=\left|E\left(a_{p-q+2},\ldots,a_p,\ 1:\rho_2,\ldots,\rho_q\cdot \frac{\rho_1 z}{a_1\cdots a_{p-q+1}}\right)\right|. \tag{10} \]

The inner integral of the function (10) is likewise expanded into a continued fraction and represented by two terms according to the right-hand side of equality (8), etc. After a \((q-1)\)-fold replacement of integrals of the form (8) by two terms and complete integration of all the terms, taking into account equalities (8) and (9), we finally obtain equality (3), as was required to prove.

1. Let us prove the theorem on the inequality of polynomials.

Theorem. If the roots of the polynomial \(Q_{2\kappa+2}(z)\) separate the roots of the polynomial \(Q_{2\kappa}(z)\), i.e.

\[ \begin{gathered} Q_{2\kappa}(z)=(z+\beta_1)\cdots(z+\beta_\kappa),\\ Q_{2\kappa+2}(z)=(z+\alpha_1)\cdots(z+\alpha_{\kappa+1}),\quad \kappa=1,2,\ldots,\\ \beta_0=\alpha_1>\beta_1>\alpha_2>\ldots>\alpha_\kappa>\beta_\kappa>\alpha_{\kappa+1}=\beta_{\kappa+1}>0, \end{gathered} \tag{11} \]

then for these polynomials the inequality

\[ |Q_{2\kappa+2}(z)|\geq |Q_{2\kappa}(z)|\,|z+\alpha|,\quad \alpha_1\cdots\alpha_{\kappa+1}=\beta_1\cdots\beta_\kappa \alpha, \]

\[ \operatorname{Re} z\geq 0,\quad \alpha_1>\alpha>\alpha_{\kappa+1}. \tag{12} \]

holds.

Proof. By direct calculation it is not difficult to prove that inequality (12) is true for two factors, i.e.

\[ |z+c|\,|z+d|\geq |z+a|\,|z+b|,\quad ab=cd,\quad 0<a\leq b<d,\quad \operatorname{Re} z\geq 0. \tag{13} \]

Let \(\beta_{l-1}>\alpha>\beta_l,\ l=1,\ldots,\kappa+1\). Then, on the basis of inequalities (11) and (12), we obtain the following inequalities:

\[ \left\{ \begin{aligned} &|z+\alpha_1|\,|z+\alpha_2| \geq |z+\beta_1|\left|z+\frac{\alpha_1\alpha_2}{\beta_1}\right|, \quad \left(\alpha_1>\frac{\alpha_1\alpha_2}{\beta_1}>\alpha_2\right);\\ &\cdots\\ &\left|z+\frac{\alpha_1\cdots\alpha_l}{\beta_1\cdots\beta_{l-1}}\right| \,|z+\alpha_{l+1}| \geq |z+\alpha|\left|z+\frac{\beta_l\cdots\beta_\kappa}{\alpha_{l+2}\cdots\alpha_{\kappa+1}}\right|,\\ &\left( \frac{\alpha_1\cdots\alpha_l}{\beta_1\cdots\beta_{l-1}} > \frac{\alpha_1\cdots\alpha_{l+1}}{\beta_1\cdots\beta_{l-1}\alpha} = \frac{\beta_l\cdots\beta_\kappa}{\alpha_{l+2}\cdots\alpha_{\kappa+1}} > \beta_l \right);\\ &\cdots\\ &\left|z+\frac{\beta_{\kappa-1}\beta_\kappa}{\alpha_{\kappa+1}}\right| \,|z+\alpha_{\kappa+1}| \geq |z+\beta_{\kappa-1}|\,|z+\beta_\kappa|,\\ &\left( \frac{\beta_{\kappa-1}\beta_\kappa}{\alpha_{\kappa+1}} > \beta_{\kappa-1}>\beta_\kappa>\alpha_{\kappa+1} \right). \end{aligned} \right. \tag{14} \]

Multiplying the left- and right-hand sides of inequalities (14) and canceling both sides of the resulting inequality by equal factors, we obtain the required inequality (12), as was to be proved.

1. For the \(E\)-function the following recurrence relation holds:

\[ E(a_1,\ldots,a_p:\rho_1,\ldots,\rho_q:z) =s_{\kappa-1}(z)+R_{\kappa-1}(z)= \tag{15} \]

\[ =s_{\kappa-1}(z)+\left(-\frac{1}{z}\right)^\kappa E(a_1+\kappa,\ldots,a_p+\kappa,\ 1:\rho_1+\kappa,\ldots,\rho_q+\kappa,\ 1+\kappa:z). \]

Relation (15) is easily verified by substituting series (1) in place of the \(E\)-function and comparing the coefficients on the left- and right-hand sides of the equality for identical powers \(z^{-n}\), where \(n=k, k+1,\ldots\). If the parameters of the function located on the right-hand side of (15) satisfy inequalities (2), then formula (3) can be applied to estimate this function; and this is equivalent to estimating the remainder term \(R_{k-1}(z)\) of the partial sum \(S_{k-1}\) of function (15).

The Lommel function ([5], p. 50 (72)) is represented by the following MacRobert function:

\[ S_{\mu,\nu}(z)= \frac{ z^{\mu-1}E\!\left(\frac{1-\nu-\mu}{2},\,\frac{1+\nu-\mu}{2},\,1::\frac{z^{2}}{4}\right) }{ \Gamma\!\left(\frac{1-\nu-\mu}{2}\right) \Gamma\!\left(\frac{1+\nu-\mu}{2}\right) }, \]

\[ -2\leq \mu<-1,\quad 0\leq \nu<1,\quad \nu+\mu<-1. \tag{16} \]

In view of equalities (3), (15), and (16), the Lommel function can be represented as the sum of a segment of series (1) and a remainder term:

\[ \begin{aligned} S_{\mu,\nu}(z) ={}&z^{\mu-1}\sigma_{\kappa-1}\!\left(\frac{z^{2}}{4}\right) +\left(\frac{4}{|z|^{2}}\right)^{\kappa}|z|^{\mu-1} \left(\frac{1-\nu-\mu}{2}\right)_{\kappa} \left(\frac{1+\nu-\mu}{2}\right)_{\kappa} \times \\ &\times\left[ \frac{1}{1+\dfrac{(2\kappa-1-\mu)^{2}-\nu^{2}}{z^{2}}} +\right.\\ &\left. +\frac{|z|^{2}(2\kappa-\mu)} {4\left(\dfrac{-1+\nu-\mu}{2}+\kappa\right)_{2} \left(\dfrac{-1-\nu-\mu}{2}+\kappa\right)_{2}} \right]. \end{aligned} \tag{17} \]

For example, in equality (17), for \(\mu=-2\), \(\nu=0\), \(z=20\), \(\kappa=9\), the remainder term \(\Delta_{9}<0.00000000015\).

REFERENCES

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3. V. E. Kornilov. Application of continued fractions to the computation of certain integrals. Izv. TPI, vol. 131, pp. 26–30, Tomsk, 1965.

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

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

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