RECURRENCE RELATIONS

FOR HYPERGEOMETRIC FUNCTIONS

V. E. Kornilov

(Presented by the Department of Higher Mathematics)

In the present paper, known identical transformations of two particular hypergeometric series are used to obtain recurrence relations between these series.

Elementary relations are obtained in the general case for the following functions:

\[ F(a,b;c;z)=\sum_{m=0}^{\infty}\frac{(a)_m(b)_m}{m!(c)_m}z^m,\quad |z|<1; \]

\[ \Phi(a;c;z)=\sum_{m=0}^{\infty}\frac{(a)_m}{m!(c)_m}z^m,\quad |z|<+\infty; \]

\[ {}_2F_0(a,b;;z)=\sum_{m=0}^{\infty}\frac{(a)_m(b)_m}{m!}z^m,\quad |z|\ll 1, \]

where

\[ (a)_m=a(a+1)\cdots(a+m-1),\quad (a)_0=1. \]

1. Let us study the following continued fraction (identity) ([1], p. 98; [2], p. 132)

\[ \left\{ \begin{aligned} \frac{F(a,b+1;c+1;z)}{F(a,b;c;z)} &= \cfrac{1}{1-\cfrac{u_1z}{1}-\cfrac{v_1z}{1}-\cdots- \cfrac{v_nz\,F(a+n,b+n+1;c+2n+1;z)} {F(a+n,b+n;c+2n;z)}} , \\[6pt] u_j&=\frac{(a+j-1)(c-b+j-1)}{(c+2j-2)_2},\quad v_j=\frac{(b+j)(c-a+j)}{(c+2j-1)_2},\quad j=1,\cdots,n. \end{aligned} \right. \tag{1} \]

Let us represent the right-hand side of identity (1) by a suitable fraction

\[ \frac{F(a,b+1;c+1;z)}{F(a,b;c;z)} = \frac{p_{2n}F_1-v_nzp_{2n-1}F_2} {q_{2n}F_1-v_nzq_{2n-1}F_2}. \]

From the last identity it is not difficult to obtain the following elementary relation

\[ \left\{ \begin{aligned} F(a,b;c;z)&=q_{2n}F(a+n,b+n;c+2n;z)-\\ &\quad -v_n z q_{2n-1}F(a+n,b+n+1;c+2n+1;z). \end{aligned} \right. \tag{2} \]

2. The following are two limiting cases of identity (1) ([2], pp. 134–137):

\[ \left\{ \begin{aligned} \frac{\Phi(a+1;c+1;z)}{\Phi(a;c;z)} &= \frac{1}{1} -\frac{a_1z}{1} -\frac{b_1z}{1} -\cdots- \frac{b_nz\,\Phi(a+n+1;c+2n+1;z)\,[\Phi(a+n;c+2n;z)]^{-1}}{1},\\ a_j&=\frac{c-a+j-1}{(c+2j-2)_2},\qquad b_j=-\frac{a+j}{(c+2j-1)_2}. \end{aligned} \right. \tag{3} \]

\[ \left\{ \begin{aligned} \frac{{}_2F_0(a,b+1;;z)}{{}_2F_0(a,b;;z)} &= \frac{1}{1} -\frac{c_1z}{1} -\frac{d_1z}{1} -\cdots-\\ &\quad -\frac{d_nz\,{}_2F_0(a+n,b+n+1;;z)\,[{}_2F_0(a+n,b+n;;z)]^{-1}}{1},\\ c_j&=a+j-1,\qquad d_j=b+j,\qquad j=1,\ldots,n. \end{aligned} \right. \tag{4} \]

3. Similarly to item 1, from equalities (3) and (4) we obtain elementary relations for the functions:

\[ \left\{ \begin{aligned} \Phi(a;c;z)&=B_{2n}\Phi(a+n;c+2n;z)-\\ &\quad -b_nz\,B_{2n-1}\Phi(a+n+1;c+2n+1;z). \end{aligned} \right. \tag{5} \]

\[ \left\{ \begin{aligned} {}_2F_0(a,b;;z)&=D_{2n}{}_2F_0(a+n,b+n;;z)-\\ &\quad -d_nz\,D_{2n-1}{}_2F_0(a+n,b+n+1;;z). \end{aligned} \right. \tag{6} \]

4. The values of the polynomials in equalities (2), (5), (6) are as follows:

\[ \left\{ \begin{aligned} q_{2n}&=q_{2n+1}+u_{n+1}zq_{2n-1},\qquad q_{2n-1}=\\ &=\sum_{m=0}^{n-1}\frac{(-1)^m C_{n-1}^m z^m}{(c+2n-1-m)_m} \left[ \sum_{k=0}^{m} \frac{(-1)^k C_m^k(m+1-2n)_k\,\lambda_m} {(1-n)_k(c)_k[(a)_k(b)_k]^{-1}} \right]. \end{aligned} \right. \tag{7} \]

\[ \left\{ \begin{aligned} B_{2n-1} &=\sum_{m=0}^{n-1} \frac{(-1)^m C_{n-1}^m z^m}{(c+2n-1-m)_m} \left[ \sum_{k=0}^{m} \frac{(-1)^k C_m^k(m+1-2n)_k} {(1-n)_k(c)_k[(a)_k]^{-1}} \right],\\ B_{2n} &=\sum_{m=0}^{n} \frac{(-1)^m C_n^m z^m}{(c+2n-m)_m} \left[ \sum_{k=0}^{m} \frac{(-1)^k C_m^k(m-2n)_k} {(-n)_k(c)_k[(a)_k]^{-1}} \right]. \end{aligned} \right. \tag{8} \]

\[ \left\{ \begin{aligned} D_{2n}&=D_{2n+1}+a_{n+1}zD_{2n-1},\qquad D_{2n-1}=\\ &=\sum_{m=0}^{n-1}\frac{C_{n-1}^m}{z^{-m}} \left[ \sum_{k=0}^{\left[\frac{m}{2}\right]} \frac{C_m^k(-1)^k(k-m)_k(a)_k(b)_k} {(1-n)_k[(a+b+n+2k-m)_{m-2k}]^{-1}} \right]. \end{aligned} \right. \tag{9} \]

In equality (7) the coefficient \(\lambda_m\) is the following:

\[ \lambda_m= \sum_{p=0}^{\left[\frac{m-k}{2}\right]} \frac{C_{m-k}^p(-1)^p(m-k-2p+1)_p(a+k)_p(b+k)_p} {(n-k-p)_p[(a+b+n+2k+2p-m)_{m-2p-2k}]^{-1}}. \tag{10} \]

References

1. H. Bateman and A. Erdélyi. Higher Transcendental Functions. Moscow, Fizmatgiz, 1965.

2. A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problems of Approximate Analysis. Moscow, Gostekhizdat, 1956.