Application of Continued Fractions to the Computation of Functions Related to Bessel Functions
Abstract Generated abstract
The paper develops continued fraction based approximations for the MacRobert E-function in the case relevant to asymptotic representations of several functions associated with Bessel functions. It proves a representation of this E-function as a finite sum of elementary fractions plus a remainder term, giving explicit formulas for the coefficients and an estimate of the remainder in the half-plane with nonnegative real part. An auxiliary inequality is established for integrals involving polynomials with negative roots, and the resulting approximation framework is connected to the Weber, Struve, and Lommel functions through their known representations in terms of the MacRobert E-function.
Full Text
APPLICATION OF CONTINUED FRACTIONS TO THE COMPUTATION OF FUNCTIONS RELATED TO BESSEL FUNCTIONS
V. E. KORNILOV
(Presented by the Department of Higher Mathematics)
The article examines the question of representing the MacRobert \(E\)-function (1) in the form of a sum of suitable fractions. An estimate is also obtained for the remainder term in modulus, valid in the complex domain \(\operatorname{Re} z > 0\).
A special case of function (1) is given by the asymptotic representations of the Bessel, Struve, and Lommel functions ([1], pp. 46–52), ([2], p. 242), which are directly related to Bessel functions (see Sec. 3).
- For the MacRobert \(E\)-function ([3], p. 201)
\[ E(a_1,a_2,1::z)=E(z)=\sum_{m=0}^{\infty}\Gamma(a_1+m)\Gamma(a_2+m)\left(-\frac{1}{z}\right)^m, \]
\[ |z|\gg 1,\quad a_1>0,\quad a_2>0,\quad \operatorname{Re}z>0, \tag{1} \]
we shall prove the theorem.
Theorem. Function (1) is represented by the following sum of suitable fractions:
\[ E(z)=\Gamma(a_1)\Gamma(a_2)\left[ \frac{n!}{(a_1+1)_n} + \sum_{m=1}^{n} \frac{a_1 b_m p_{2n+1}(z:a_m)}{a_m q_{2n+1}(z:a_m)} + R_{2n+1}(z) \right], \quad n=1,2,\ldots,\ (a)_n=a(a+1)\ldots(a+n-1), \tag{2} \]
\[ p_{2n+1}(z)= \sum_{\kappa=0}^{n} z^n \sum_{m=0}^{\,n-\kappa} C_n^\kappa \frac{(-1)^{\,n-\kappa-m}}{(n+a_2-\kappa-m)_{\kappa+1}}. \]
The coefficients \(a_m\) and \(b_m\) in equality (2) have positive values [4], pp. 5, 24, and, in view of [5], p. 28, (7), are determined according to the equalities
\[ \prod_{m=1}^{n}\left(1+\frac{z}{a_m}\right) = \sum_{k=0}^{n} C_k^n \frac{z^k}{(a_1+1)_k} = q_{2n+1}(z), \]
\[ a_1>a_2>\ldots>a_n>0, \tag{3} \]
\[ b_m=-P_{2n+1}(-a_m):\left\{a_m\frac{d}{dz}\left[q_{2n+1}(-a_m)\right]\right\}, \]
\[ P_{2n+1}(z)= \sum_{\kappa=0}^{n-1} z^{\kappa+1} \sum_{m=0}^{\,n-\kappa-1} C_n^m \frac{(-1)^{\,n-\kappa-m-1}}{(n+a_1-\kappa-m)_{\kappa+1}}. \tag{4} \]
The modulus of the remainder term of equality (2) is less than the following sum:
\[ \begin{aligned} |R_{2n+1}(z)|<{}& \frac{n!(a_2)_2}{(a_1+1)_n(a_2+2)_2} \left\{\prod_{m=1}^{n}\left[1+\frac{x^2+y^2}{a_m^2(a_2+2)_2}\right]\right\}^{-1} \\ &\quad+ \sum_{m=1}^{n} \frac{q_m a_1 n!}{a_m(a_2+1)_n\,|q_{2n+1}(z:a_m)q_{2n+3}(z:a_m)|}, \end{aligned} \tag{5} \]
\[ \operatorname{Re} z \ge 0,\qquad n=1,2,\ldots \]
Proof. We represent the function (1) by a double integral [3], p. 201, (3), and, in view of ([6], p. 144), [5] p. 28 and [7], p. 374, (67), replace the inner integral by the sum of a convergent fraction and the remainder term; we represent the convergent fraction as a sum of elementary fractions
\[ \begin{aligned} \left\{ E(z)={}& \int_{0}^{\infty} e^{-v}v^{a_2-1}\,dv \int_{0}^{\infty} e^{-u}u^{a_1-1}(1+uv:z)^{-1}\,du \\ ={}& \Gamma(a_1)\int_{0}^{\infty} e^{-v}v^{a_2-1} \left[ \frac{p_{2n+1}\!\left(\frac{z}{v}\right)} {q_{2n+1}\!\left(\frac{z}{v}\right)} +r_{2n+1}\!\left(\frac{z}{v}\right) \right]\,dv \\ ={}& \Gamma(a_1)\left[ \sum_{m=1}^{n} b_m \times \frac{a_1}{a_m} \int_{0}^{\infty} \frac{e^{-v}v^{a_2-1}\,dv}{1+a_m v:z} + \frac{\Gamma(a_2)n!}{(a_1+1)_n} + \rho_{2n+1}\!\left(\frac{z}{v}\right) \right] \\ ={}& \Gamma(a_1)\Gamma(a_2)\times \left\{ \sum_{m=1}^{n}\frac{a_1b_m}{a_m} \left[ \frac{p_{2n+1}\!\left(\frac{z}{a_m}\right)} {q_{2n+1}\!\left(\frac{z}{a_m}\right)} + r_{2n+1}\!\left(\frac{z}{a_m}\right) \right] + \frac{n!}{(a_1+1)_n} + \frac{1}{\Gamma(a_2)}\rho_{2n+1}\!\left(\frac{z}{v}\right) \right\}. \end{aligned} \tag{6} \]
In view of [5] (pp. 27–28) we have
\[ \left|r_{2n+1}\!\left(\frac{z}{a_m}\right)\right| = \left| \sum_{k=n}^{\infty} \frac{k!\,a_2}{(a_2+1)_{k+1}q_{2k+1}q_{2k+3}} \right| < \frac{a_2 n!}{|q_{2n+1}q_{2n+3}|} \times \]
\[ \times \sum_{k=0}^{\infty} \frac{(n+1)_k}{(a_2+1)_{k+n+1}} \frac{n!\,[a_2+1)_n]}{|q_{2n+1}q_{2n+3}|}; \qquad \operatorname{Re} z \ge 0,\quad q_k=q_k\!\left(\frac{z}{a_m}\right). \tag{7} \]
Further, in view of equalities (3), (6), and inequality (7), we have
\[ \left| \varepsilon_{2n+1}\!\left(\frac{z}{v}\right) \right| < \frac{n!}{(a_1+1)_n} \int_{0}^{\infty} e^{-v}v^{a_2-1} \left[ \prod_{m=1}^{n} \left(1+\frac{x^2+y^2}{v^2a_m^2}\right) \right]^{-1} \,dv = \]
\[ = \frac{n!}{(a_1+1)_n} \sum_{m=1}^{n} c_m \int_{0}^{\infty} e^{-v}v^{a_2-1} \left(1+\frac{x^2+y^2}{v^2a_m^2}\right)^{-1} \,dv = \varepsilon_{2n+1}(x,y). \tag{8} \]
We represent each of the integrals on the right-hand side of equality (8) by an infinite series, applying the formula [3], p. 19, (15), and, in view of equality (1), the first equality (6), and inequality (11), we have
\[ \varepsilon_{2n+1}(x,y) = \frac{n!\,2^{a_2+1}}{\sqrt{\pi}(a_1+1)_n} \sum_{m=1}^{n} \frac{c_m a_m^2}{x^2+y^2} \int_{0}^{\infty} e^{-v}v^{\frac{a_2}{2}}v\,dv \times \]
\[ \times \int_0^\infty e^{-u}u^{\frac{\alpha_2}{2}+\frac12} \left(1+\frac{4a_n^2uv}{x^2+y^2}\right)^{-1}du \leq \]
\[ \leq \frac{\Gamma(\alpha_2)(\alpha_2)_2}{(\alpha_1+1)_n(\alpha_2+2)_2} \left\{ \prod_{m=1}^n \left[ 1+\frac{x^2+y^2}{a_n^2(\alpha_2+2)_2} \right] \right\}^{-1}, \qquad \operatorname{Re} z>0. \tag{9} \]
On the basis of equality (6), inequalities (7) and (9), the theorem is proved.
- Let us prove an inequality for one integral.
Theorem. If the polynomial \(Q_{4n}(x)\) has negative roots
\[ Q_{4n}(x)=\prod_{m=1}^{2n}\left(1+\frac{x}{\beta_m}\right), \qquad n=1,2,\ldots, \tag{10} \]
\[ \beta_1>\beta_2>\cdots>\beta_{2n}>0, \]
then for the following integral \(J\) the inequality
\[ J=\int_0^\infty \frac{e^{-v}v^{\alpha-1}\,dv}{Q_{4n}(x:v)} \leq \frac{\Gamma(\alpha)}{Q_{4n}(x:\alpha)}, \qquad \alpha>0,\ x>0 \tag{11} \]
is valid.
Proof. We represent the quotient \(1:Q_{4n}(x:v)\) by elementary fractions
\[ \frac{1}{Q_{4n}(x:v)} = 1+\sum_{k=1}^n \frac{b_kx}{x+v\beta_{2k-1}} -\sum_{k=1}^n \frac{c_kx}{x+v\beta_{2k}}, \qquad \beta_{2k}<\beta_{2k-1}. \tag{12} \]
On the basis of equalities (11) and (12) we have
\[ \begin{aligned} J &= \frac{\Gamma(\alpha)}{Q_{4n}\!\left(\dfrac{x}{\alpha}\right)} +\Gamma(\alpha) \Biggl\{ \sum_{k=1}^n b_k \left[ \lim_{m\to\infty} \frac{p_{2m}\!\left(\dfrac{x}{\beta_{2k-1}}\right)} {q_{2m}\!\left(\dfrac{x}{\beta_{2k-1}}\right)} - \frac{x}{x+\beta_{2k-1}\alpha} \right] \\ &\qquad\qquad -\sum_{k=1}^n c_k \left[ \lim_{m\to\infty} \frac{p_{2m}(x:\beta_{2k})} {q_{2m}(x:\beta_{2k})} - \frac{x}{x+\beta_{2k}\alpha} \right] \Biggr\} \leq \frac{\Gamma(\alpha)}{Q_{4n}(x:\alpha)}. \end{aligned} \tag{13} \]
In view of [6], p. 12, and equality (12), we obtain the inequalities
\[ \frac{x}{x+\beta_{2k-1}\alpha} \leq \frac{x}{x+\beta_{2k-1}(\alpha-\varepsilon)} \leq \frac{p_{2n+2}(x:\beta_{2k-1})}{q_{2n+2}(x:\beta_{2k-1})}, \tag{14} \]
\[ n=1,2,\ldots;\qquad \alpha>0,\ \varepsilon>0,\ \varepsilon\to0, \]
\[ \frac{x}{x+\beta_{2k-1}\alpha} \leq \frac{x}{x+\beta_{2k}\alpha} \leq \frac{x}{x+\beta_{2k}(\alpha-\varepsilon)} \leq \frac{p_{2n+2}(x:\beta_{2k})}{q_{2n+2}(x:\beta_{2k})}, \tag{15} \]
\[ n=1,2,\ldots. \]
Further, in view of inequalities (14) and (15), the sum enclosed in braces in the left-hand side of inequality (13) satisfies the inequality
\[ \sum_{k=1}^n b_k \left[ \lim_{m\to\infty} \frac{p_{2m}(x:\beta_{2k-1})}{q_{2m}(x:\beta_{2k-1})} - \frac{x}{x+\beta_{2k-1}\alpha} \right] - \sum_{k=1}^n c_k \left[ -\frac{x}{x+\beta_{2k}\alpha} + \right. \]
\[ +\lim_{m\to\infty}\frac{p_{2m}(x:\beta_{2k})}{q_{2m}(x:\beta_{2k})} \leq \sum_{\kappa=1}^{n} b_{\kappa} \left[ \frac{x}{x+\beta_{2\kappa-1}(a-\varepsilon)} - \frac{x}{x+\beta_{2\kappa-1}a} \right] - \]
\[ -\sum_{\kappa=1}^{n} c_{\kappa} \left[ \frac{x}{x+\beta_{2\kappa}(a-\varepsilon)} - \frac{x}{x+\beta_{2\kappa}a} \right] =F(x). \tag{16} \]
The right-hand side of inequality (16), in view of inequality (12), has a nonpositive value
\[ F(x)= \frac{1}{Q_{4n}\left(\dfrac{x}{a-\varepsilon}\right)} - \frac{1}{Q_{4n}\left(\dfrac{x}{a}\right)} \leq 0, \tag{17} \]
therefore, on the basis of inequalities (16) and (17), inequalities (13) and (11) are also valid, as was required to prove.
The inequality of the form (11) for a polynomial of odd degree is proved analogously.
- The asymptotic representations of the Weber function \(\Omega_\nu(z)\), the Struve function \(S_\nu(z)\) [2], p. 242, and the Lommel function \(S_{\mu,\nu}(z)\) [1], p. 50, (72), are expressed by means of function (1)
(\(Y_\nu(z)\) is the Bessel function of the second kind)
\[ \Omega_\nu(z)+Y_\nu(z)= \frac{1-\cos \nu\pi}{\pi z}\cdot \frac{ E\left(\dfrac12-\dfrac{\nu}{2},\,\dfrac12+\dfrac{\nu}{2},\,1::\dfrac{z^2}{4}\right) }{ \Gamma\left(\dfrac12-\dfrac{\nu}{2}\right) \Gamma\left(\dfrac12+\dfrac{\nu}{2}\right) } - \]
\[ - \frac{1-\cos \nu\pi}{\pi z}\cdot \frac{ \nu E\left(1-\dfrac{\nu}{2},\,1+\dfrac{\nu}{2},\,1::\dfrac{z^2}{4}\right) }{ z\Gamma\left(1-\dfrac{\nu}{2}\right) \Gamma\left(1+\dfrac{\nu}{2}\right) }; \qquad -1<\nu<1. \tag{18} \]
\[ S_\nu(z)=Y_\nu(z)+ \frac{ E\left(\dfrac12,\,\dfrac12-\nu,\,1::\dfrac{z^2}{4}\right) }{ \pi\left(\dfrac{z}{2}\right)^{1-\nu} \Gamma\left(\dfrac12+\nu\right) \Gamma\left(\dfrac12-\nu\right) }, \qquad \nu<\frac12. \tag{19} \]
\[ S_{\mu,\nu}(z)= \frac{ E\left(\dfrac{1-\nu-\mu}{2},\,\dfrac{1+\nu-\mu}{2},\,::\dfrac{z^2}{4}\right) }{ z^{1-\mu} \Gamma\left(\dfrac{1-\nu-\mu}{2}\right) \Gamma\left(\dfrac{1+\nu-\mu}{2}\right) }, \tag{20} \]
\[ \mu+\nu<1,\qquad \mu-\nu<1. \]
REFERENCES
-
H. Bateman and A. Erdélyi. Higher Transcendental Functions (Bessel functions, parabolic cylinder functions, orthogonal polynomials). Moscow, Fizmatgiz, 1966.
-
E. Jahnke and F. Emde. Tables of Functions with Formulae and Curves. Moscow, Fizmatgiz, 1959.
-
H. Bateman and A. Erdélyi. Higher Transcendental Functions (hypergeometric function, Legendre functions). Moscow, Fizmatgiz, 1965.
-
T. I. Stieltjes. Investigations on Continued Fractions. Moscow, ONTI, 1963.
-
V. E. Kornilov. Application of continued fractions to the computation of certain kinds of integrals. Izv. TPI, vol. 131, pp. 26–30, Tomsk, 1965.
-
A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Questions of Approximate Analysis. Moscow, Gostekhizdat, 1956.
-
V. I. Smirnov. A Course of Higher Mathematics, vol. III, part II. Moscow, Gostekhizdat, 1953.