Application of the Product of Binomials to the Computation of Kummer’s Asymptotic Series
Abstract Generated abstract
This paper develops approximations to Kummer’s asymptotic series by representing it as a finite product of binomials, intended for computation when the argument has large modulus. Using continued fraction expansions for ratios of related Kummer series, the author derives polynomial recurrences whose roots determine the binomial factors and exponents in the product representation. For parameters satisfying negative one less than beta less than or equal to alpha and zero less than alpha less than two, and for complex arguments with argument bounded by pi over two in modulus, the paper establishes estimates for the remainder term. The result is a constructive approximation formula together with explicit bounds controlling its error in the stated sector.
Full Text
APPLICATION OF THE PRODUCT OF BINOMIALS FOR CALCULATING KUMMER’S ASYMPTOTIC SERIES
V. E. KORNILOV
(Presented by the scientific seminar of the Department of Higher Mathematics)
Approximations of Kummer’s asymptotic series in the form of a product of binomials are obtained on the basis of the theory of continued fractions and can be used to compute the indicated series for large values of the modulus of the argument.
Kummer’s series contains the parameters $\alpha$ and $\beta$, which in this article are studied as varying in the following intervals: $-1 < \beta \leq \alpha$, $0 < \alpha < 2$. An estimate of the remainder term is obtained for the complex variable $z$ in the region $|\arg z| \leq \dfrac{\pi}{2}$.
- If we introduce the abbreviated notation for the product
\[ (a)_k=a(a+1)\ldots(a+k-1)=\frac{\Gamma(a+k)}{\Gamma(a)}, \qquad (a)_0=1, \]
then the asymptotic series of the Kummer function ([1], p. 341) is the following:
\[ F\left(\alpha,\beta;\,;-\frac{1}{z}\right) = \sum_{k=0}^{\infty} \frac{(\alpha)_k(\beta)_k}{(1)_k} \left(-\frac{1}{z}\right)^k, \qquad |z|\gg 1,\quad |\arg z|<\pi . \tag{1} \]
Series (1) is transformed in the following way:
\[ F\left(\alpha,\beta;\,;-\frac{1}{z}\right) = e^{\ln F\left(\alpha,\beta;\,;-\frac{1}{z}\right)} = e^{\ln F\left(-\frac{1}{z}\right)} = \]
\[ = \exp\left\{ \int_{\infty}^{z} \left[ F\left(-\frac{1}{z}\right) \right]^{-1} d\left[ F\left(-\frac{1}{z}\right) \right] \right\}, \]
that is,
\[ F\left(\alpha,\beta;\,;-\frac{1}{z}\right) = \exp\left[ \int_{\infty}^{z} \frac{ \alpha\beta F\left(\alpha+1,\beta+1;\,;-\frac{1}{t}\right) }{ F\left(\alpha,\beta;\,;-\frac{1}{t}\right) } \cdot \frac{dt}{t^2} \right]. \tag{2} \]
Next, to the numerator of the integrand of function (2) we apply the recurrence relation
\[ \beta F\left(\alpha+1,\beta+1;\,;-\frac{1}{t}\right) = tF\left(\alpha,\beta;\,;-\frac{1}{t}\right) - tF\left(\alpha+1,\beta;\,;-\frac{1}{t}\right). \tag{3} \]
and replace the ratio of the two series by its known expansion as a continued fraction ([2], p. 137); then series (1) is expressed by the number \(e\), whose exponent contains an integral and, under the integral sign, a continued fraction, namely:
\[ F\left(a,\beta;\,;\,-\frac{1}{z}\right) = \exp\left\{ \int_{\infty}^{z} \left[ \frac{a}{t}-\frac{aP_{2n+1}(t)}{tQ_{2n+1}(t)} \right]dt \right\} \times \exp\left[ a\int_{\infty}^{z}R_{2n+1}(t)\frac{dt}{t} \right], \tag{4} \]
where
\[ r_{2n+1}(z)=a\int_{\infty}^{z}R_{2n+1}(t)\frac{dt}{t}; \tag{5} \]
\[ \frac{P_k(t)}{Q_k(t)} = \frac{1}{a_1+\dfrac{1}{a_2+\ddots+\dfrac{1}{a_k}}}, \]
\[ a_1=1,\qquad a_{2m}=\frac{(a+1)_{m-1}t}{(\beta)_m},\qquad a_{2m+1}=\frac{(\beta)_m}{(a+1)_m}; \tag{6} \]
\[ R_{2n+1}(t)= \sum_{\kappa=n+1}^{\infty} \frac{(\beta)_\kappa} {(a+1)_\kappa Q_{2\kappa-1}(t)Q_{2\kappa+1}(t)} \quad ([13], p. 34). \tag{7} \]
For the numerator and denominator of the convergent fraction (6), the following equalities and relation are known ([3], pp. 13, 14):
\[ P_{2n+1}(t)=1+\cdots+\frac{t^n}{(a+1)_n} = \frac{t^n}{(a+1)_n}p_{2n+1}(t), \tag{8} \]
\[ Q_{2n+1}(t)= \sum_{m=0}^{n}a_{2m+1} +\cdots+ \frac{n(a+\beta+n)}{(a+1)_n}t^{\,n-1} + \]
\[ +\frac{t^n}{(a+1)_n} = \frac{t^n}{(a+1)_n}q_{2n+1}(t), \tag{9} \]
where
\[ D_n=\sum_{m=0}^{n}a_{2m+1} = \sum_{m=0}^{n}\frac{(\beta)_m}{(a+1)_m}. \tag{10} \]
\[ Q_{2n+1}(t)= \left( a_{2n}a_{2n+1}+1+\frac{a_{2n+1}}{a_{2n-1}} \right)Q_{2n-1}(t) - \frac{a_{2n+1}}{a_{2n-1}}Q_{2n-3}(t). \tag{11} \]
- The roots of the polynomial \(Q_{2n+1}(t)\) are real negative and are separated by the roots of \(Q_{2n-1}(t)\) ([3], p. 14). In view of equality (9), it is not difficult to establish that the extreme root of \(Q_{2n+1}(t)\) is located in the interval \((-\alpha-\beta-n,\,-n\alpha-n\beta-n^2)\); therefore, for relatively adjacent denominators the inequality
\[ \left| 1+\frac{t}{n(\alpha+\beta+n)} \right| \frac{D_n}{D_{n-1}} \left|Q_{2n-1}(t)\right| < \left|Q_{2n+1}(t)\right|, \qquad |\arg t|\leq \frac{\pi}{2}, \]
\[ n=1,2,\ldots \tag{12} \]
is valid. Inequality (12) also holds for \(-1<\beta<0\), since \(Q_{2n+1}(t)\) contains the parameters \(\alpha\) and \(\beta\) in symmetric form and under interchange
intervals of variation of these parameters all incomplete convergents of the continued fraction (6) are positive.
Academician V. I. Smirnov proved the formula for the hypergeometric series ([4], p. 374):
\[ F(\alpha,\beta;\gamma;1)= \frac{\Gamma(\gamma)\Gamma(\gamma-\alpha-\beta)} {\Gamma(\gamma-\alpha)\Gamma(\gamma-\beta)}. \tag{13} \]
According to equality (10),
\[ D_n=F(1,\beta;\alpha+1;1) - F(1,\beta+n+1;\alpha+n+2;1)\, \frac{(\beta)_{n+1}}{(\alpha+1)_{n+1}} \tag{14} \]
and, in view of formula (13), we obtain:
\[ D_n=\sum_{m=0}^{n}\frac{(\beta)_m}{(\alpha+1)_m} = \frac{(\alpha)_{n+1}-(\beta)_{n+1}} {(\alpha+1)_n(\alpha-\beta)}; \quad \alpha\ne\beta. \tag{15} \]
Applying formulas (13), (15) and inequality (12) to the infinite functional series (7), taking into account that
\[ \frac{\alpha}{\alpha-\beta}<D_n<1,\quad n=1,2,\ldots;\quad -1<\beta<0. \quad D_n<D_{n+1}\ \text{for}\ \beta>0; \]
\[ D_n>\frac{1}{2}[C+\ln(n+1)]+\ln 2> \frac{1}{2}\ln 7\,(n+1)\quad ([5], p. 244), \]
\[ \beta=\alpha>\frac{1}{2}, \tag{16} \]
we obtain
\[ |R_{2n+1}(t)|< S_n\, \frac{|(\beta)_{n+1}|\,D_{n-1}D_n^{-1}} {(\alpha+1)_{n+1}|Q_{2n+1}(t)|\,|Q_{2n+3}(t)|}, \quad |\arg t|\leq \frac{\pi}{2}, \tag{17} \]
where
\[ S_n= \begin{cases} \displaystyle D_n^2\,\frac{(\alpha+n+1)(\alpha-\beta)}{\alpha^2}, & \text{for } -1<\beta<0;\\[1.2em] \displaystyle \frac{D_n(\alpha+n+1)}{D_{n+1}(\alpha-\beta)}, & \text{for } 0<\beta<\alpha;\\[1.2em] \displaystyle \frac{4D_n^2\,[1+(\alpha+n+1)\ln 7\,(n+2)]} {\ln 7\,(n+1)\,\ln 7\,(n+2)}, & \text{for } \beta=\alpha>\frac{1}{2}. \end{cases} \tag{18} \]
Next, with the aid of inequalities (12) and (17), we obtain an estimate for the modulus of the integral (5), where the integration is performed along the radius of the point \(z\) ([6], p. 430), namely:
\[ |\rho_{2n+1}(z)|< \frac{\alpha S_n |(\beta)_{n+1}|(n+1)(\alpha+\beta+n+1)} {(\alpha+1)_{n+1}} \int_{\infty}^{z} \frac{|dt|} {|Q_{2n+1}(t)|^2|t|^2} < \]
\[ < \frac{\alpha S_n |(\beta)_{n+1}|(n+1)(\alpha+\beta+n+1)} {(\alpha+1)_{n+1}|Q_{2n+1}(z)|^2} \int_{\infty}^{|z|} \frac{d|t|}{|t|^2} = \]
\[ = \frac{\alpha S_n |(\beta)_{n+1}|(n+1)(\alpha+\beta+n+1)} {(\alpha+1)_{n+1}|Q_{2n+1}(z)|^2|z|}. \]
The last inequality is valid because, for \(|\arg t|\leq \frac{\pi}{2}\), the polynomials in the denominators of the integrands can be rep—
are composed by a product of binomials of the form \(\left(1+\dfrac{t}{a_m}\right)\), \(a_m>0\), and the modulus of each of them assumes its minimum value at the upper limit of the integral. Thus an estimate of the integral (5) in modulus has been obtained:
\[ |\rho_{2n+1}(z)|< \frac{\alpha S_n|(\beta)_{n+1}|(n+1)(\alpha+\beta+n+1)} {(\alpha+1)_{n+1}|Q_{2n+1}(z)|\,|z|}, \qquad |\arg z|\leqslant \frac{\pi}{2}, \tag{19} \]
where \(S_n\) assumes values according to equality (18).
- For \(n=1\) we obtain an estimate of the function (4):
\[ \left|F\left(\alpha,\beta;;-\frac{1}{z}\right)\right| \leq \left|1+\frac{\alpha+\beta+1}{z}\right|^{-\frac{\alpha\beta}{\alpha+\beta+1}} \exp |\rho_3(z)|. \tag{20} \]
Equality (4), taking (8) and (9) into account, can be written in the form of the following sum:
\[ F\left(\alpha,\beta;;-\frac{1}{z}\right) = \exp\left\{\int_\infty^z \left[\frac{\alpha}{t}-\frac{\alpha p_{2n+1}(t)}{tq_{2n+1}(t)}\right]dt \right\} + \]
\[ +\, r_{2n+1}(z)=F_{2n+1}(z)+r_{2n+1}(z), \tag{21} \]
where
\[ r_{2n+1}(z)=F_{2n+1}(z)\{\exp[\rho_{2n+1}(z)]-1\}. \tag{22} \]
In view of (20) and (22) we obtain
\[ |r_{2n+1}(z)|< \left|1+\frac{\alpha+\beta+1}{z}\right|^{-\frac{\alpha\beta}{\alpha+\beta+1}} \exp|\rho_3(z)|\,[\exp|\rho_{2n+1}(z)|-1]. \tag{23} \]
Theorem. The series (1) is represented by the following product of binomials \((n=1,2,\ldots)\):
\[ F\left(\alpha,\beta;;-\frac{1}{z}\right) = \prod_{m=1}^{n}\left(1+\frac{a_m}{z}\right)^{b_m} + r_{2n+1}(z), \tag{24} \]
where
\[ \prod_{m=1}^{n}\left(1+\frac{a_m}{z}\right)=q_{2n+1}(z), \qquad b_m= \frac{\alpha\beta A_{2n+1}(-a_m)} {a_m^2\dfrac{d}{dz}[q_{2n+1}(-a_m)]}. \tag{25} \]
The polynomials \(A_{2n+1}(z)\) and \(q_{2n+1}(z)\) are the following:
\[ A_1(z)=0,\qquad A_3(z)=q_1(z)=1;\qquad q_3(z)=1+\frac{\alpha+\beta+1}{z}, \tag{26} \]
and for \(n=2,3,\ldots\) they are determined successively by means of the relations:
\[ q_{2n+1}(z)= \left[1+(\alpha+\beta+2n-1)\frac{1}{z}\right]q_{2n-1}(z) - \]
\[ -(\alpha+n-1)(\beta+n-1)\frac{1}{z^2}q_{2n-3}(z). \tag{27} \]
The upper bound for \(|r_{2n+1}(z)|\) in the region \(|\arg z|\leqslant \dfrac{\pi}{2}\) is established with the aid of inequalities (23), (19).
Proof. First of all, we introduce the relation
\[ \beta A_{2n+1}(z)=z\,[q_{2n+1}(z)-p_{2n+1}(z)];\quad n=1,2,\ldots \tag{28} \]
and, in view of equalities (8), (9), (11), and (28), the polynomials \(q_{2n+1}(z)\), \(p_{2n+1}(z)\), and \(A_{2n+1}(z)\) satisfy relation (27).
Applying equality (28), we transform equality (21):
\[ F\left(\alpha,\beta;\,;\,-\frac{1}{z}\right) = \exp\left[ -\int_{\infty}^{z} \frac{\alpha\beta A_{2n+1}(t)}{q_{2n+1}(t)} \,d\left(\frac{1}{t}\right) \right]+r_{2n+1}(z) = \]
\[ = \exp\left[ \sum_{m=1}^{n} a_m b_m \int_{\infty}^{z} \frac{1}{\left(1+\frac{a_m}{t}\right)} \,d\left(\frac{1}{t}\right) \right]+r_{2n+1}(z) = \]
\[ = \prod_{m=1}^{n} \left(1+\frac{a_m}{z}\right)^{b_m} +r_{2n+1}(z), \]
where
\[ b_m= \frac{\alpha\beta A_{2n+1}(-a_m)} {a_m^2\,\dfrac{d}{dz}\,[q_{2n+1}(-a_m)]}, \]
moreover, on the basis of (24) and (21), \(|r_{2n+1}(z)|\) is estimated by inequalities (23), (19).
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