Transformation of Some Power Series into Continued Fractions
Abstract Generated abstract
The paper derives recurrence transformations that express selected hypergeometric and confluent hypergeometric power series in forms suitable for expansion as continued fractions. It treats the Gauss series with parameters involving an integer n, the Kummer series in analogous integer-parameter cases, and an asymptotic Kummer-type series for large z with negative real part. By applying known recurrence relations and elementary coefficient comparisons, the author represents the original series as sums of finite polynomial terms plus a remaining standard series, such as F with second parameter 1 or Phi with first parameter 1, whose continued fraction expansions are already known. The resulting formulas provide explicit reductions for these series to continued fraction representations, including related transformations obtained from standard identities for hypergeometric and Kummer functions.
Full Text
TRANSFORMATION INTO CONTINUED FRACTIONS OF CERTAIN POWER SERIES
V. E. KORNILOV
(Presented by the scientific seminar of the Department of Higher Mathematics)
In the article recurrence relations are obtained for the following series:
1) the sum of the Gauss series \(F(\alpha,\beta;\gamma;z)\), with parameter \(\beta=n\), \(\gamma-n\), is transformed to the series \(F(\alpha+n-1,1;\gamma;z)\);
2) the sum of the Kummer series \(\Phi(\alpha;\gamma;z)\), in the case \(\alpha=n\), \(\gamma-n\), is transformed to the sum of the series \(\Phi(1;\gamma;z)\);
3) the asymptotic series of the Kummer function
\[ F\left(\alpha,\beta;;\frac{1}{z}\right) = \sum_{\kappa=0}^{\infty} \frac{(\alpha)_{\kappa}(\beta)_{\kappa}}{(1)_{\kappa}z^{\kappa}}, \qquad |z|\gg 1,\quad Re\,(z)<0 \tag{1} \]
in the case \(\beta=n\) is transformed to the sum of the series \(F\left(\alpha+n-1,1;;\frac{1}{z}\right)\).
The power series obtained after the transformations are expanded into continued fractions, which can be found in [1] and [2]. The initial series, however, are represented by the sum of two polynomials, one of these sums being multiplied by the sum of an infinite series that is expanded into a continued fraction.
In the article the following abbreviated notation for the product is introduced
\[ (a)_k=a(a+1)\ldots(a+k-1)=\frac{\Gamma(a+k)}{\Gamma(a)},\qquad (a)_0=1. \]
- The power series
\[ F(a,1;\gamma;z)=\sum_{n=0}^{\infty}\frac{(a)_n}{(\gamma)_n}z^n,\quad |z|<1,\quad \gamma\ne 0,-1,\ldots, \tag{2} \]
which is obtained by expanding into a series a certain integral with a variable upper limit ([3], p. 308):
\[ F(a,1;\gamma;z)= \frac{(\gamma-1)z^{1-\gamma}}{(1-z)^{a+1-\gamma}} \int_{0}^{z} z^{\gamma-2}(1-z)^{a-\gamma}\,dz, \qquad \gamma>1, \tag{3} \]
can be expanded into a continued fraction ([1], p. 320)
\[ F(\alpha,1;\gamma;z)= \cfrac{1}{1-\cfrac{\alpha z}{\gamma-\ddots-\cfrac{k(\gamma-\alpha+k-1)z}{\gamma+2k-1-\cfrac{(\gamma+k-1)(\alpha+k)z}{\gamma+2k-\ddots}}}} \tag{4} \]
and satisfies the relation
\[ F(\alpha,1;\gamma;z)=-(\gamma-1)\sum_{n=0}^{k-1} \frac{(\alpha)_n}{(\alpha-\gamma+1)_{n+1}}(1-z)^n+ \]
\[ +\frac{(\alpha)_k}{(\alpha-\gamma+1)_k}(1-z)^k F(\alpha+k,1;\gamma;z). \tag{5} \]
The sum of the series (2) satisfies the relation
\[ F(\alpha,1;\gamma;z)= -\frac{\gamma-1}{\alpha-\gamma+1} +\frac{\alpha}{\alpha-\gamma+1}(1-z)F(\alpha+1,1;\gamma;z), \tag{6} \]
since the coefficient of \(z^k\) on the right-hand side of the equality is
\[ \frac{\alpha}{\alpha-\gamma+1} \left[ \frac{(\alpha+1)_k}{(\gamma)_k} - \frac{(\alpha+1)_{k-1}}{(\gamma)_{k-1}} \right] = \]
\[ = \frac{\alpha(\alpha+1)_{k-1}\bigl[(\alpha+k)-(\gamma+k-1)\bigr]} {(\alpha-\gamma+1)(\gamma)_k} = \frac{(\alpha)_k}{(\gamma)_k}. \]
Equality (5) for \(k=1\) coincides with relation (6). Suppose that formula (5) is valid; then, replacing \(F(\alpha+k,1;\gamma;z)\) by formula (6), we obtain:
\[ F(\alpha,1;\gamma;z)=-(\gamma-1)\sum_{n=0}^{k-1} \frac{(\alpha)_n}{(\alpha-\gamma+1)_{n+1}}(1-z)^n+ \]
\[ +\frac{(\alpha)_k}{(\alpha-\gamma+1)_k}(1-z)^k \left[ -\frac{\gamma-1}{\alpha-\gamma+k+1} +\right. \]
\[ \left. +\frac{\alpha+k}{\alpha-\gamma+k+1} (1-z)F(\alpha+k+1,1;\gamma;z) \right] = \]
\[ =-(\gamma-1)\sum_{n=0}^{k} \frac{(\alpha)_n}{(\alpha-\gamma+1)_{n+1}}(1-z)^n+ \]
\[ +\frac{(\alpha)_{k+1}}{(\alpha-\gamma+1)_{k+1}} (1-z)^{k+1}F(\alpha+k+1,1;\gamma;z), \]
which confirms relation (5).
Analogously one may consider the properties of the sums of the series
\[ \Phi(1;\gamma;z)=\sum_{n=0}^{\infty}\frac{z^n}{(\gamma)_n}, \qquad \gamma\ne 0,-1,\ldots, \tag{7} \]
\[ F\left(\alpha,1;;\frac{1}{z}\right) = \sum_{n=0}^{\infty}\frac{(\alpha)_n}{z^n}, \qquad |z|\gg 1,\quad \operatorname{Re}(z)<0, \tag{8} \]
which can be transformed respectively to the incomplete gamma function and to Tricomi’s function ([4], p. 110), whose expansions into continued fractions are known ([2], p. 301), namely:
\[ \Phi(1;\gamma;z)= \cfrac{1}{ 1-\cfrac{z}{ \gamma-\cfrac{\kappa z}{ \gamma+2\kappa-1-\cfrac{(\gamma+\kappa-1)z}{\gamma+2\kappa-\cdots}}}} \tag{9} \]
\[ F\left(\alpha,1;;\frac{1}{z}\right)= \cfrac{1}{ 1+\cfrac{\alpha}{ -z+\cfrac{\kappa}{ 1+\cfrac{\alpha+\kappa}{-z+\cdots}}}} \tag{10} \]
The sum of the power series (7) satisfies the relation:
\[ \Phi(1;\gamma;z)= \sum_{n=0}^{\kappa-1}\frac{z^n}{(\gamma)_n} + \frac{z^\kappa}{(\gamma)_\kappa}\, \Phi(1;\gamma+\kappa;z). \tag{11} \]
The sum of the asymptotic series (8) satisfies the relation:
\[ F\left(\alpha,1;;\frac{1}{z}\right)= \sum_{n=0}^{\kappa-1}\frac{(\alpha)_n}{z^n} + \frac{(\alpha)_\kappa}{z^\kappa}\, F\left(\alpha+\kappa,1;;\frac{1}{z}\right). \tag{12} \]
Equalities (11), (12) are easily verified by substituting the series (7), (8) and comparing the coefficients of equal powers of \(z\) in the left- and right-hand sides of the equalities under consideration.
- If, to the following function ([3], p. 308):
\[ F(\alpha,n;\gamma;z)= \tag{13} \]
\[ = \sum_{\kappa=1}^{n} (-1)^{\kappa+1}C_{n-1}^{\kappa-1} \frac{(\alpha+1-\kappa)_{n-1}}{(1)_{n-1}}\, F(\alpha+n-\kappa,1;\gamma;z), \]
which stands on the right-hand side of the equality, we apply equality (5), then we obtain
\[ F(\alpha,n;\gamma;z)= \]
\[ = \sum_{\kappa=1}^{n} (-1)^{\kappa+1}C_{n-1}^{\kappa-1} \frac{(\alpha+1-\kappa)_{n-1}}{(1)_{n-1}} \left[ -(\gamma-1) \sum_{m=0}^{\kappa-2} \frac{(\alpha+n-\kappa)_m}{(\alpha-\gamma+n-\kappa+1)_{m+1}}\, (1-z)^m \right. \]
\[ \left. + \frac{(\alpha+n-\kappa)_{\kappa-1}} {(\alpha-\gamma+n-\kappa+1)_{\kappa-1}}\, (1-z)^{\kappa-1}F(\alpha+n-1,1;\gamma;z) \right]. \]
After expanding the brackets we obtain:
\[ F(\alpha,n;\gamma;z)= \]
\[ = \frac{\gamma-1}{(1)_{n-1}} \sum_{\kappa=2}^{n} (-1)^\kappa C_{n-1}^{\kappa-1} \sum_{m=0}^{\kappa-2} \frac{(\alpha+1-\kappa)_{n+m-1}} {(\alpha-\gamma+n-\kappa+1)_{m+1}}\, (1-z)^m + \tag{14} \]
\[ + \frac{(\alpha)_{n-1}}{(1)_{n-1}} \sum_{\kappa=1}^{n} \frac{(1-n)_{\kappa-1}(1-\alpha)_{\kappa-1}} {(1)_{\kappa-1}(1+\gamma-\alpha-n)_{\kappa-1}}\, (1-z)^{\kappa-1}F(\alpha+n-1,1;\gamma;z). \]
The series \(F(\alpha+n-1,1;\gamma;z)\) can now be represented by the continued fraction (4). The hypergeometric series \(F(\alpha,\gamma-n;\gamma;z)\), in view of the known relation ([5], p. 319)
\[ F(\alpha,\gamma-n;\gamma;z)=(1-z)^{n-\alpha}F(\gamma-\alpha,n;\gamma;z) \tag{15} \]
and formula (14), can also be expressed by means of the continued fraction (4).
- After successive application of the known recurrence relation ([5], p. 334)
\[ \Phi(n;\gamma;z)=\frac{\gamma-1}{n-1}\Phi(n-1;\gamma-1;z) -\frac{\gamma-n}{n-1}\Phi(n-1;\gamma;z) \tag{16} \]
one obtains
\[ \begin{aligned} \Phi(n;\gamma;z) &= \\ &=\sum_{\kappa=0}^{n-1}(-1)^{\kappa}C_{n-1}^{\kappa} \frac{(\gamma-n)_n}{(1)_{n-1}(\gamma-n+\kappa)} \Phi(1;\gamma-n+1+\kappa;z). \end{aligned} \tag{17} \]
To the function situated on the right-hand side of equality (17), equality (11) is applied:
\[ \begin{aligned} \Phi(n;\gamma;z) &=\frac{(\gamma-n)_n}{(1)_{n-1}} \sum_{\kappa=0}^{n-1}C_{n-1}^{\kappa} \frac{(-1)^{\kappa}}{\gamma-n+\kappa} \left[ \sum_{m=0}^{n-\kappa-2} \frac{z^m}{(\gamma-n+1+\kappa)_m} \right.\\ &\qquad\left. +\frac{z^{\,n-\kappa-1}}{(\gamma-n+1+\kappa)_{n-\kappa-1}} \Phi(1;\gamma;z) \right]. \end{aligned} \]
Expanding the brackets, we finally obtain
\[ \begin{aligned} \Phi(n;\gamma;z) &=\frac{(\gamma-n)_n}{(1)_{n-1}} \sum_{\kappa=0}^{n-2}C_{n-1}^{\kappa}(-1)^{\kappa} \sum_{m=0}^{n-\kappa-2} \frac{z^m}{(\gamma-n+\kappa)_{m+1}} + \\ &\quad +\frac{1}{(1)_{n-1}} \sum_{\kappa=0}^{n-1} C_{n-1}^{\kappa}(-1)^{\kappa} (\gamma-n)_{\kappa}z^{\,n-\kappa-1} \Phi(1;\gamma;z), \end{aligned} \tag{18} \]
where \(\Phi(1;\gamma;z)\) is expanded into the continued fraction (9). The series \(\Phi(\gamma-n;\gamma;-z)\), in view of the known relation ([5], p. 334)
\[ \Phi(\gamma-n;\gamma;-z)=e^{-z}\Phi(n;\gamma;z) \tag{19} \]
is also represented by the continued fraction (9).
- The sum of the asymptotic series \(F\left(\alpha,n;;\dfrac{1}{z}\right)\) satisfies the recurrence relation
\[ (n-1)F\left(\alpha,n;;\frac{1}{z}\right) =\alpha F\left(\alpha+1,n-1;;\frac{1}{z}\right)- \]
\[ -(\alpha-n+1)F\left(\alpha,n-1;;\frac{1}{z}\right), \tag{20} \]
since the coefficient of \(z^{-k}\) on the right-hand side of the equality is
\[ \frac{(\alpha)_{k+1}(n-1)_k}{(1)_k} -\frac{(\alpha-n+1)(\alpha)_k(n-1)_k}{(1)_k} = \frac{(\alpha)_k(n-1)_{k+1}}{(1)_k}, \]
which proves the validity of this formula.
After successive application of relation (20), we obtain
\[ F\left(a,n;;\frac{1}{z}\right) = \sum_{\kappa=0}^{n-1} (-1)^{\kappa} C_{n-1}^{\kappa} \frac{(a-\kappa)_{n-1}}{(1)_{n-1}} F\left(a+n-\kappa-1;1;;\frac{1}{z}\right). \tag{21} \]
Applying equality (12), we have
\[ \begin{aligned} F\left(a,n;;\frac{1}{z}\right) &= \sum_{\kappa=0}^{n-1} (-1)^{\kappa} C_{n-1}^{\kappa} \frac{(a-\kappa)_{n-1}}{(1)_{n-1}} \left[ \sum_{m=0}^{\kappa-1} \frac{(a+n-\kappa-1)_m}{z^m} +\right.\\ &\qquad\qquad\qquad\left. +\frac{(a+n-\kappa-1)_{\kappa}}{z^{\kappa}} F\left(a+n-1,\,1;;\frac{1}{z}\right) \right]. \end{aligned} \]
After expanding the brackets, we get:
\[ \begin{aligned} F\left(a,n;;\frac{1}{z}\right) &= \frac{1}{(1)_{n-1}} \sum_{\kappa=1}^{n-1} C_{n-1}^{\kappa}(-1)^{\kappa} \sum_{m=0}^{\kappa-1} \frac{(a-\kappa)_{n+m-1}}{z^m} +\\ &\quad+ \frac{(a)_{n-1}}{(1)_{n-1}} \sum_{\kappa=0}^{n-1} C_{n-1}^{\kappa} \frac{(1-a)_{\kappa}}{z^{\kappa}} F\left(a+n-1,1;;\frac{1}{z}\right), \end{aligned} \tag{22} \]
where the asymptotic series \(F\left(a+n-1;1;;\frac{1}{z}\right)\) can be represented by the continued fraction (10).
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