Application of Continued Fractions to the Computation of Incomplete Gauss and Kummer Functions
Abstract Generated abstract
The paper studies the computation of incomplete Gauss and Kummer functions, defined by replacing a constant integration limit in the standard integral representations with a variable limit. It derives series representations for the resulting component integrals and uses positivity of associated Hankel determinants to represent each series as the sum of a convergent continued fraction and a remainder term. The remainder is shown, in the specified domains of the complex variable, to have modulus smaller than the difference between two adjacent convergents. Parameter restrictions are identified to ensure the required monotonicity conditions for the hypergeometric factors in limiting cases.
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APPLICATION OF CONTINUED FRACTIONS TO THE COMPUTATION OF INCOMPLETE GAUSS AND KUMMER FUNCTIONS
V. E. KORNILOV
(Presented by the Department of Higher Mathematics)
The paper considers the question of representing incomplete Gauss and Kummer functions by the sum of a suitable fraction and a remainder term. In this case, the modulus of the remainder term is smaller than the modulus of the difference of two adjacent suitable fractions in the corresponding domain of the complex variable \(z\).
- The Gauss function \(F\) and the Kummer function \({}_2F_0\) (asymptotic series) are represented by the following integrals:
\[ \frac{\Gamma(\gamma-\beta)\Gamma(\beta)}{\Gamma(\gamma)} F(\alpha,\beta;\gamma;z)=J_1+J_2= \int_0^1 t^{\beta-1}(1-t)^{\gamma-\beta-1}(1-tz)^{-\alpha}\,dt; \quad \beta>0,\ \gamma>\beta. \tag{1} \]
\[ \Gamma(\alpha){}_2F_0\left(\alpha,\beta;\frac{1}{z}\right) =I_1+I_2= \int_0^\infty t^{\alpha-1}e^{-t}(1-t:z)^{-\beta}\,dt; \quad \alpha>0. \tag{2} \]
By analogy with incomplete beta- and gamma-functions, we shall call, respectively, the integrals (1) or (2) incomplete Gauss or Kummer functions if one of the constant limits of integration is replaced by a variable limit of integration \(x\) \((0<x<1\) or \(0<x<+\infty)\).
- The integrals (1) and (2), with the aid of the variable limit \(x\), will be represented respectively as sums of the integrals \(J_1, J_2\) and \(I_1, I_2\); we transform the resulting integrals and finally obtain:
\[ \left\{ \begin{aligned} x^{-\beta}J_1 &=\sum_{n=0}^{\infty} a_n z^n =\sum_{n=0}^{\infty} \frac{(\alpha)_n}{(\beta+n)\cdot n!} F(\beta+n;\beta+1-\gamma;\beta+n+1;x)\times \\ &\quad \times x^n z^n;\quad 0<x<1,\ \gamma>\beta+1,\ \alpha<1. \end{aligned} \right. \tag{3} \]
\[ \left\{ \begin{aligned} \frac{(1-z)^\alpha}{(1-x)^{\gamma-\beta}}J_2 &=\sum_{n=0}^{\infty} b_n z^n =\sum_{n=0}^{\infty} \frac{(\alpha)_n}{(\gamma-\beta+n)\cdot n!}\times \\ &\quad \times F(\gamma-\beta+n,1-\beta;\gamma-\beta+n+1;1-x) (1-x)^n z^n (z-1)^{-n}; \\ &\quad 0<x<1,\ \beta>1,\ \alpha<1. \end{aligned} \right. \tag{4} \]
\[ \left\{ \begin{aligned} x^{-\alpha} I_1 &= \sum_{n=0}^{\infty} c_n z^{-n} = \sum_{n=0}^{\infty} \frac{(\beta)_n}{n!} \left[\int_{0}^{1} t^{\alpha+n-1} e^{-xt}\,dt\right] \left(\frac{x}{z}\right)^n;\\ 0 &< x < +\infty,\quad \beta < 1. \end{aligned} \right. \tag{5} \]
\[ \left\{ \begin{aligned} \frac{e^x x^{1-\alpha}}{(1-x:z)^{-\beta}} I_2 &= \sum_{n=0}^{\infty} d_n \frac{1}{(z-x)^n} \\ &= \sum_{n=0}^{\infty} \frac{(\beta)_n}{n!} \left[\int_{0}^{\infty} e^{-t} t^n \left(1+\frac{t}{x}\right)^{\alpha-1} dt\right] (z-x)^{-n};\\ 0 &< x < +\infty,\quad \beta < 1. \end{aligned} \right. \tag{6} \]
On the basis of equalities (1)—(6), the theorem ([1], p. 211), and also ([2], p. 226; [3], problem 1220), the determinants
\[ \left| \begin{array}{ccccc} a_p & \cdots & a_{p+m-1}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ a_{p+m-1} & \cdots & a_{p+2m-2} \end{array} \right|; \ldots; \left| \begin{array}{ccccc} d_p & \cdots & d_{p+m-1}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ d_{p+m-1} & \cdots & d_{p+2m-2} \end{array} \right|; \]
\[ p,\ m=1,\ 2,\ldots \]
are positive. Therefore the power series (3)—(6) are each represented by the sum of a convergent fraction and a remainder term which, in absolute value, is smaller than the absolute value of the difference of two neighboring convergents; and these estimates are valid, respectively, in the following domains of the complex variable \(z\) [4]:
\[ \operatorname{Re} z,\quad \operatorname{Re}\left(\frac{z}{z-1}\right),\quad \operatorname{Re}\left(\frac{1}{z}\right),\quad \operatorname{Re}(z-x)^{-1} \le 0. \]
When establishing the limits of variation of the parameters of the functions \(F\) in equalities (3), (4), it is necessary to choose the parameter values so that, in the limiting case (\(x=1\) or \(x=0\), respectively), these functions decrease monotonically as the index \(n\) increases [4].
For example, for the function \(F\) of series (3) we have ([5], p. 374, (67))
\[ F(\beta+n,\ \beta+1-\gamma;\ \beta+n+1;\ 1) = \frac{\Gamma(\beta+n+1)\Gamma(\gamma-\beta)} {\Gamma(1)\Gamma(\gamma+n)}. \]
The sequence
\[ \left\{\frac{\Gamma(\beta+n+1)}{\Gamma(\gamma+n)}\right\},\quad n=0,\ 1,\ 2,\ldots \]
decreases monotonically only in the case when \(\gamma>\beta+1\). For formula (4), this condition is the inequality \(\beta>1\).
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