Elementary Relations Between Partial Hypergeometric Series
Abstract Generated abstract
The paper derives elementary identities connecting shifted partial hypergeometric series by applying known finite continued fraction transformations for quotients of such series. It treats the Gauss hypergeometric function, the confluent hypergeometric function, the divergent hypergeometric series of type 2F0, and the function I, with special attention to the case b = 0 except for I, where the general case is considered. From these continued fraction identities and their limiting cases, the author obtains finite relations expressing functions with initial parameters in terms of shifted functions multiplied by explicitly given polynomial coefficients. The paper also records formulas and limiting procedures for the coefficient polynomials appearing in these relations.
Full Text
ELEMENTARY RELATIONS BETWEEN PARTIAL HYPERGEOMETRIC SERIES
V. E. KORNILOV
(Presented by the Department of Higher Mathematics)
In this article, known identical transformations of a quotient of two hypergeometric series are used to obtain elementary relations between these series.
For the functions \(F\), \(\Phi\), \({}_2F_0\), elementary relations were obtained in the special case \((b=0)\), and for the function \(I\) in the general case (here \(F \equiv F(a,b;c;z)\), etc.). In the article the following notation is used for the product
\[
(a)_k=a(a+1)\ldots(a+k-1).
\]
The relations are obtained for the following series:
\[
F(a,b;c;z)=\sum_{m=0}^{\infty}\frac{(a)_m(b)_m}{m!(c)_m}z^m,\quad |z|<1;\quad \Phi(b;c;z)=
\]
\[
=\sum_{m=0}^{\infty}\frac{(b)_m z^m}{m!(c)_m};\quad {}_2F_0(a;b;;z)=\sum_{m=0}^{\infty}\frac{(a)_m(b)_m}{m!}z^{-m},
\]
\[
|z|\ll1;\quad I(c,z)=\sum_{m=0}^{\infty}\frac{z^m}{m!(c)_m}.
\]
1. Let us study the following finite continued fraction (identity) ([1], p. 98; [2], p. 132):
\[
\left\{
\begin{aligned}
\frac{F(a,b+1;c+1;z)}{F(a,b;c;z)}
&\equiv
\frac{1}{1-\dfrac{u_1z}{1}-\dfrac{v_1z}{1}-\cdots-
\dfrac{v_nz\,\dfrac{F(a+n,b+n+1;c+2n+1;z)}
{F(a+n,b+n;c+2n;z)}}{1}},\\[6pt]
u_j&=\frac{(a+j-1)(c-b+j-1)}{(c+2n-2)_2},\\[6pt]
v_j&=\frac{(b+j)(c-a+j)}{(c+2j-1)_2},\quad j=1,\ldots,n.
\end{aligned}
\right.
\tag{1}
\]
Representing the right-hand side of identity (1) by a suitable fraction in the special case \((b=0)\), we obtain
\[
\frac{F(a,1;c+1;z)}{1}
\equiv
\frac{p_{2n}F_1-v_nzp_{2n-1}F_2}{q_{2n}F_1-v_nzq_{2n-1}F_2}.
\]
(explicit values of \(p_m, q_m\) in item 4). From the last identity it is not difficult to obtain the following elementary relations:
\[ \begin{cases} F(a,1;\ c+1;\ z)=p_{2n}F(a+n,n;\ c+2n;\ z)-\\ \quad -p_{2n-1}v_n zF(a+n,n+1;\ c+2n-1;\ z); \end{cases} \tag{2} \]
\[ \begin{cases} 1=q_{2n}F(a+n,n;\ c+2n;\ z)-\\ \quad -v_n zq_{2n-1}F(a+n,n+1;\ c+2n-1;\ z). \end{cases} \tag{3} \]
2. Three limiting cases of identity (1) are the following ([2], pp. 134–137):
\[ \begin{cases} \dfrac{\Phi(b+1;\ c+1;\ z)}{\Phi(b;\ c;\ z)} \equiv \dfrac{1}{1}-\dfrac{a_1z}{1}-\dfrac{b_1z}{1}-\cdots-\\[6pt] \quad -\dfrac{b_nz\Phi(b+n+1;\ c+2n+1;\ z)[\Phi(b+n;\ c+2n;\ z)]^{-1}}{1},\\[8pt] a_j=\dfrac{c-b+j-1}{(c+2j-2)_2},\quad b_j=\dfrac{-b-j}{(c+2j-1)_2},\quad j=1,\ldots,n; \end{cases} \tag{4} \]
\[ \begin{cases} \dfrac{{}_2F_0(a,b+1;;z)}{{}_2F_0(a,b;;z)} \equiv \dfrac{1}{1}-\dfrac{c_1z}{1}-\dfrac{d_1z}{1}-\cdots-\\[6pt] \quad -\dfrac{d_nz\,{}_2F_0(a+n,b+n+1;;z)[{}_2F_0(a+n,b+n;;z)]^{-1}}{1},\\[8pt] c_j=a+j-1,\quad d_j=b+j,\quad j=1,\ldots,n; \end{cases} \tag{5} \]
\[ \begin{cases} I(c+1,z)[I(c,z)]^{-1} \equiv \dfrac{1}{1}+\dfrac{e_1z}{1}+\dfrac{g_1z}{1}+\cdots\\[6pt] \quad \cdots+\dfrac{g_nzI(c+2n+1,z)[I(c+2n,z)]^{-1}}{1},\\[8pt] e_j^{-1}=(c+2j-2)_2,\quad g_j^{-1}=(c+2j-1)_2. \end{cases} \tag{6} \]
3. Analogously to item 1, we obtain elementary relations for the partial values of functions from equalities (4), (5) (\(b=0\)) and functions \(I\):
\[ \begin{cases} \Phi(1;\ c+1;\ z)=A_{2n}\Phi(n;\ c+2n;\ z)+\\ \quad +b_nzA_{2n-1}\Phi(n+1;\ c+2n+1;\ z); \end{cases} \tag{7} \]
\[ \begin{cases} 1=B_{2n}\Phi(n;\ c+2n;\ z)+\\ \quad +b_nzB_{2n-1}\Phi(n+1;\ c+2n+1;\ z); \end{cases} \tag{8} \]
\[ \begin{cases} {}_2F_0(a,1;;z)=C_{2n}{}_2F_0(a+n,n;;z)-\\ \quad -d_nzC_{2n-1}{}_2F_0(a+n,n+1;;z); \end{cases} \tag{9} \]
\[ \begin{cases} 1=D_{2n}{}_2F_0(a+n,n;;z)-\\ \quad -d_nzD_{2n-1}{}_2F_0(a+n,n+1;;z); \end{cases} \tag{10} \]
\[ \begin{cases} I(c+1,z)=E_{2n}I(c+2n,z)+\\ \quad +g_nzE_{2n-1}I(c+2n+1,z); \end{cases} \tag{11} \]
\[ \begin{cases} I(c,z)=G_{2n}I(c+2n,z)+\\ \quad +g_{2n}zG_{2n-1}I(c+2n+1,z). \end{cases} \tag{12} \]
- The values of the polynomials in equalities (2), (3), (7)—(12) are the following ([3], pp. 21, 23; [4]):
\[ \left\{ \begin{aligned} q_{2n-1} & = \sum_{k=0}^{n-1} C_{n-1}^{k} \frac{(1-a-n)_{k}}{(2-c-2n)_{k}}(-z)^{k}; \\[4pt] q_{2n} & = \sum_{k=0}^{n} C_{n}^{k} \frac{(1-a-n)_{k}}{(1-c-2n)_{k}}(-z)^{k}; \\[4pt] p_{2n-1} & = \sum_{k=0}^{n-1} C_{n-1}^{k} \frac{(1-a-n)_{k}}{(2-c-2n)_{k}}(-z)^{k} \sum_{m=0}^{\,n-k-1}\frac{(a+1)_{m}}{(c+2)_{m}}z^{m}; \\[4pt] p_{2n} & = \sum_{k=0}^{n-1} C_{n}^{k} \frac{(1-a-n)_{k}}{(1-c-2n)_{k}}(-z)^{k} \sum_{m=0}^{\,n-k-1}\frac{(a)_{m}}{(c+1)_{m}}z^{m}. \end{aligned} \right. \tag{13} \]
\[ \left\{ \begin{aligned} A_m & = \lim_{a\to\infty} p_m\left(\frac{z}{a}\right); \qquad B_m = \lim_{a\to\infty} q_m\left(\frac{z}{a}\right); \\[4pt] C_m & = \lim_{c\to\infty} p_m(cz); \qquad D_m = \lim_{c\to\infty} q_m(cz). \end{aligned} \right. \tag{14} \]
\[ \left\{ \begin{aligned} E_{2n-1} & = \sum_{k=0}^{n-1} \frac{C_{2n-k-2}^{k} z^{k}} {(c+1)_{k}(c+2n-1-k)_{k}}; \\[4pt] E_{2n} & = \sum_{k=0}^{n-1} C_{2n-k-1}^{k}\bigl[(c+1)_{k}(c+2n-k)_{k}\bigr]^{-1} z^{k}; \\[4pt] G_{2n-1} & = \sum_{k=0}^{n-1} C_{2n-k-1}^{k}\bigl[(c)_{k}(c+2n-1-k)_{k}\bigr]^{-1} z^{k}; \\[4pt] G_{2n} & = \sum_{k=0}^{n} C_{2n-k}^{k}\bigl[(c)_{k}(c+2n-k)_{k}\bigr]^{-1} z^{k}; \\[4pt] n & = 1,2,\ldots \end{aligned} \right. \tag{15} \]
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