Application of Continued Fractions to the Computation of Generalized Hypergeometric Functions

V. E. Kornilov

(Presented by the Department of Higher Mathematics of the Tomsk Polytechnic Institute)

In the present article, for the generalized hypergeometric functions (22) and (23), a theorem is proved stating that they can be represented by the continued fraction (2) with positive elements of the links. If the indicated functions are computed by means of convergents, then, for these approximations, estimates of the remainders in modulus are valid according to the theorem of Sec. 2.

1. Let a power series be given

\[ f(z)=\sum_{n=0}^{\infty} c_n z^n;\quad |z|<1\ \text{or}\ |z|\ll 1, \tag{1} \]

The continued fraction corresponding to this series has positive elements of the links \((a_m,\theta_m>0,\ m=1,2,\ldots)\)

\[ f(z)= \cfrac{a_1}{ \theta_1+\cfrac{-a_2 z}{ \theta_2+\cdots+\cfrac{-a_n z}{\theta_n+\cdots}}} \tag{2} \]

In what follows the following finite continued fraction is also needed:

\[ \frac{P_k^n(z)}{Q_k^n(z)} = \theta_n+ \cfrac{-a_{n+1}z}{ \theta_{n+1}+\cdots+\cfrac{-a_{n+k}z}{\theta_{n+k}}} \tag{3} \]

The denominators of the convergents for the continued fraction (2) are as follows:

\[ \begin{cases} Q_{2n}(z)=\theta_1\ldots\theta_{2n}+\cdots+a_2a_4\ldots a_{2n}(-z)^n,\\ Q_{2n+1}(z)=\theta_1\ldots\theta_{2n+1}+\cdots+(\theta_1a_3a_5\ldots a_{2n+1}+\cdots\\ \qquad\qquad\qquad\qquad\qquad\quad \cdots+a_2a_4\ldots a_{2n}\theta_{2n+1})(-z)^n. \end{cases} \tag{4} \]

On the basis of (1), (2), and ([1], pp. 5, 24), all roots of the polynomial \(Q_m(z)\) lie in the interval \((1,+\infty)\), if the series (1) converges for \(|z|<1\), and in the interval \((0,+\infty)\) for \(|z|\ll 1\).

1. Represent the function (1) by the following sum:

\[ f(z)=\frac{P_m(z)}{Q_m(z)}+R_m(z),\quad m=1,2,\ldots \tag{5} \]

Theorem. For the modulus \(R_m(z)\) of equality (5), the inequalities

\[ |R_m(z)|<|S_\kappa|<|S_{\kappa-1}|<\cdots<|S_1|,\qquad \operatorname{Re} z\leq 0, \tag{6} \]

hold, where

\[ (-1)^m S_\kappa=\frac{P_{m+2\kappa-1}(z)}{Q_{m+2\kappa-1}(z)}-\frac{P_m(z)}{Q_m(z)},\qquad \kappa=1,2,\ldots \tag{7} \]

For the series (1), convergent in the unit disk, we have:

\[ \begin{cases} |R_m(z)|< \dfrac{a_1\ldots a_{m+1}|z|^m\lambda|\beta:z-1|} {|Q_m(z)Q_{m+1}(z)|\bigl(|\beta:z-1|-1\bigr)}, \qquad 0<\operatorname{Re}z\leq 1,\\[1.2em] |\beta:z-1|>1;\qquad \beta=\min\dfrac{\beta_{m+n}\beta_{m+n+1}}{a_{m+n+1}}, \qquad n=1,2,\ldots,\\[1.2em] \lambda=\max \dfrac{Q_2^{\,m}(1)\ldots Q_2^{\,m+n-1}(1)\,Q_m(1)Q_{m+1}(1)} {Q_{m+n}(1)Q_{m+n+1}(1)}, \qquad m=1,2,\ldots \end{cases} \tag{8} \]

Proof. To prove inequalities (6), it is sufficient to prove the inequality

\[ |S_\kappa|<|S_{\kappa-1}|,\qquad \kappa=2,3,\ldots . \tag{9} \]

On the basis of formulas ([2], p. 450, (3.4), (3.5)), we compute the difference between the convergents \(S_\kappa\)

\[ S_\kappa= \frac{a_1\ldots a_{m+1}z^m Q_{2\kappa-2}^{\,m+1}(z)} {Q_m(z)Q_{m+2\kappa-1}(z)},\qquad \kappa=1,2,\ldots . \tag{10} \]

Inequality (9), on the basis of equality (10), is transformed into the following inequality

\[ |N(z)|=|Q_{m+2\kappa-1}(z)Q_{2\kappa-4}^{\,m+1}(z)| > |Q_{m+2\kappa-3}(z)Q_{2\kappa-2}^{\,m+1}(z)|=|L(z)|. \tag{11} \]

T. J. Stieltjes, in his monograph [1], studied in detail the polynomials of inequality (11); he established the following:

1). The positive roots of the polynomials \(Q_{m+2\kappa-3}(z)\) and \(Q_{2\kappa-4}^{\,m+1}(z)\) are located, respectively, between the roots of the polynomials \(Q_{m+2\kappa-1}(z)\) and \(Q_{2\kappa-2}^{\,m+1}(z)\).

2). All roots of the polynomial \(L(z)\) are located between the extreme roots of the polynomial \(N(z)\).

We shall prove inequality (11) in the case \(m=2n+1\). On the basis of the first equality (4) and equalities (3) and (11), we have

\[ \begin{cases} N(z)=C\displaystyle\prod_{e=1}^{n+2\kappa-2}\left(1-\frac{z}{\alpha_e}\right), \qquad \alpha_1<\cdots<\alpha_{n+2\kappa-2},\\[1.2em] C=\beta_1\ldots \beta_{2n+2\kappa}\beta_{2n+3}\ldots \beta_{2n+2\kappa-2}; \end{cases} \tag{12} \]

\[ L(z)=C\prod_{e=1}^{n+2\kappa-2}\left(1-\frac{z}{\beta_e}\right), \qquad \beta_1<\cdots<\beta_{n+2\kappa-2}. \tag{13} \]

From equalities (12) and (13) it is seen that the free coefficients of the polynomials \(N(z)\) and \(L(z)\) are identical; moreover, in view of the first equality (4), we have

\[ \alpha_1\ldots \alpha_{n+2\kappa-2} = \beta_1\ldots \beta_{n+2\kappa-2}. \tag{14} \]

Next we prove that the roots of the polynomial \(N(z)L(z)\), in view of inequalities (12), (13) and ([1], pp. 10–20), satisfy the inequalities

\[ \alpha_1<\beta_1<(\alpha_2,\beta_2)<(\alpha_3,\beta_3)<\cdots \]

\[ \cdots<(\alpha_{n+2\kappa-3},\beta_{n+2\kappa-3})<\beta_{n+2\kappa-2}<\alpha_{n+2\kappa-2}, \tag{15} \]

where the inequality \(\beta_1<(\alpha_2,\beta_2)\) is equivalent to \(\beta_1<\alpha_2<\beta_2\), or \(\beta_1<\beta_2<\alpha_2\), etc. Suppose that the sequence of roots (15) is violated and is represented in the general case by the following inequalities

\[ \alpha_{m-1}<\beta_{m-1}<\alpha_m<\alpha_{m+1}<\beta_m<\beta_{m+1}<\beta_{m+2}<\alpha_{m+2}, \]
\[ m=2,\ldots,n+2\kappa-4. \tag{16} \]

Let \(\alpha_m\) belong to the polynomial \(Q_{2n+2\kappa}(z)\) and be its \(\kappa\)-th root \((\kappa\leqslant m)\). Then the root \(\alpha_{m+1}\) can belong only to the polynomial \(Q_{2\kappa-4}^{2n+2}(z)\), and it will be its \(n+1-\kappa\)-th root in the order of increasing roots. Further, we have: \(\kappa\) roots of the polynomial \(Q_{2n+2\kappa}(z)\) separate \(\kappa-1\) roots of the polynomial \(Q_{2n+2\kappa-2}(z)\); \(m-\kappa+1\) roots of the polynomial \(Q_{2\kappa-4}^{2n+2}(z)\) must follow the \(m-\kappa+1\) roots of the polynomial \(Q_{2\kappa-2}^{2n+2}(z)\), since they also separate them. Consequently, in all we have \(m\) roots \(\beta_i\) which precede the root \(\alpha_{m+1}\), whereas according to the sequence (16) we have only \(m-1\) roots \(\beta_i\) preceding the root \(\alpha_{m+1}\). On the basis of this contradiction we conclude that a sequence of roots of the form (16) cannot occur, and the roots of the polynomial \(N(z)L(z)\) are arranged according to the inequalities (15). Construct the polynomial \(M(z)\), whose roots follow every two roots of the sequence (15):

\[ \left\{ \begin{aligned} M(z)&=C\prod_{\ell=1}^{\,n+2\kappa-3}\left(1-\frac{z}{\gamma_\ell}\right), \qquad \alpha_1<\gamma_1<\alpha_2<\gamma_2<\cdots \\ &\cdots<\alpha_{n+2\kappa-2};\qquad \beta_1<\gamma_1<\beta_2<\gamma_2<\cdots<\beta_{n+2\kappa-2}. \end{aligned} \right. \tag{17} \]

We multiply the polynomial \(M(z)\) by such a binomial \(1-z:\gamma\) that the equality

\[ \gamma_1\cdots\gamma_{n+2\kappa-3}\gamma = \alpha_1\cdots\alpha_{n+2\kappa-2}; \qquad \beta_1<\gamma<\beta_{n+2\kappa-2}. \tag{18} \]

is fulfilled. Let the numbers \(\delta_1\) and \(\delta_2\) be such that \(\beta_{n+2\kappa-2}<\delta_2<\alpha_{n+2\kappa-2}\), \(\alpha_1<\delta_1<\beta_1\); then, multiplying the following inequalities ([3], 18)

\[ \left|1-\frac{z}{\gamma_1}\right| \left|1-\frac{z}{\gamma}\right| \leqslant \left|1-\frac{z}{\delta_1}\right| \left|1-\frac{\delta_1 z}{\gamma\gamma_1}\right|, \]

\[ \left|1-\frac{\delta_1 z}{\gamma\gamma_1}\right| \left|1-\frac{z}{\gamma_{n+2\kappa-3}}\right| \leqslant \left|1-\frac{z}{\delta_2}\right| \left|1-\frac{\delta_1\delta_2 z}{\gamma\gamma_1\gamma_{n+2\kappa-3}}\right|, \qquad \operatorname{Re}z\leqslant0; \]

we obtain

\[ \left|1-\frac{z}{\gamma_1}\right| \left|1-\frac{z}{\gamma}\right| \left|1-\frac{z}{\gamma_{n+2\kappa-3}}\right| \leqslant \left|1-\frac{z}{\delta_1}\right| \left|1-\frac{z}{\delta_2}\right| \left|1-\frac{\delta_1\delta_2 z}{\gamma\gamma_1\gamma_{n+2\kappa-3}}\right|, \tag{19} \]

\[ \alpha_1<\frac{\delta_1\delta_2}{\gamma\gamma_1\gamma_{n+2\kappa-3}}<\alpha_{n+2\kappa-2}, \qquad \operatorname{Re}z\leqslant0. \]

Replacing three binomials of the polynomial \((1-z:\gamma)M(z)\) by the right-hand side of inequality (19), we obtain a polynomial \(M_1(z)\), which, in view of ([3], (18)), satisfies the inequalities

\[ |L(z)|<|M_1(z)|<|N(z)|, \qquad \operatorname{Re}z\leqslant0. \tag{20} \]

On the basis of inequalities (20), inequalities (11) and (9) are valid, which is what was required to prove.

For the region \(0<\operatorname{Re}z\leqslant1\), in view of formulas (8), (11)—(15), and ([3], (18), we obtain

\[ |R_m(z)| < \frac{a_1\cdots a_{m+1}|z|^m}{|Q_m(z)Q_{m+1}(z)|} \left( 1+ \frac{a_{m+2}|z|\,Q_2^0(1)\,Q_m(1)} {|Q_2^m(z)|\,Q_{m+2}(1)} +\cdots \right) < \]

\[ < \frac{a_1\cdots a_{m+1}|z|^m\lambda}{|Q_m(z)Q_{m+1}(z)|} \sum_{n=0}^{\infty}\frac{1}{|\beta z^{-1}-1|^n}, \qquad 0<\operatorname{Re}z\leqslant1. \]

On the basis of the inequality obtained, the relations (8) are valid in this case as well.

The theorem is proved analogously in the case where in inequality (11) \(m=2n\); only in this case, in view of the second equality (4), we obtain

\[ \alpha_1 \ldots \alpha_{n+2k-2}<\beta_1 \ldots \beta_{n+2k-2}. \tag{21} \]

On the basis of equality (14), inequality (21) only strengthens inequality (11), since the roots of the polynomials \(N(z)\) and \(L(z)\) are located in the denominators of the second terms of the binomials.

1. For the generalized hypergeometric functions ([14], pp. 199, 202):

\[ \left\{ \begin{aligned} {}_{p+1}F_p(a_1,\ldots,a_p,1;\,\beta_1,\ldots,\beta_p;\,z) &=\sum_{n=0}^{\infty} d_n z^n \\ &=\sum_{n=0}^{\infty} \frac{(a_1)_n\ldots(a_p)_n}{(\beta_1)_n\ldots(\beta_p)_n}z^n; \quad |z|<1,\ (a)_0=1,\\ (a)_n&=a(a+1)\ldots(a+n-1),\quad a_i>0,\quad i=1,\ldots,p;\\ &\beta_1>a_1,\ldots,\beta_p>a_p; \end{aligned} \right. \tag{22} \]

\[ \left\{ \begin{aligned} E(a_1,\ldots,a_p,1:\rho_1,\ldots,\rho_q:z) &=\sum_{n=0}^{\infty} h_n z^n \\ &=\sum_{n=0}^{\infty} \frac{(\alpha_1)_n\ldots(\alpha_p)_n}{(\rho_1)_n\ldots(\rho_q)_n} \left(-\frac{1}{z}\right)^n; \quad |z|\gg 1,\quad p>q;\\ \alpha_i&>0,\quad i=1,\ldots,p;\quad \rho_1>\alpha_1,\ldots,\rho_q>\alpha_q; \end{aligned} \right. \tag{23} \]

the following theorem is valid.

Theorem. The continued fractions (2) corresponding to the series (22) and (23) have positive link terms

\[ a_m,\ \delta_m>0,\quad m=1,2,\ldots \]

Proof. To the power series

\[ F(a_i,1;\beta_i;z)=\sum_{n=0}^{\infty}\frac{(a_i)_n}{(\beta_i)_n}z^n =\sum_{n=0}^{\infty}c_n z^n,\quad i=1,\ldots,p \tag{24} \]

there corresponds the continued fraction (2) with positive link terms ([15], p. 21, (1)).

On the basis of ([1], p. 23), the quadratic form

\[ \sum_{i,\kappa=0}^{m-1} c_{p+i+\kappa}x_i x_\kappa \tag{25} \]

is a definite and positive form. By virtue of equalities (22)—(25), ([6], problem 1220) and ([7], p. 140, (11.5)), the quadratic forms

\[ \sum_{i,\kappa=0}^{m-1} d_{p+i+\kappa}x_i x_\kappa,\qquad \sum_{i,\kappa=0}^{m-1} h_{p+i+\kappa}x_i x_\kappa \tag{26} \]

are definite and positive forms.

In view of (26) and ([1], p. 23), the determinants

\[ \left| \begin{matrix} d_p & \ldots & d_{p+m-1}\\ \ldots & \ldots & \ldots\\ d_{p+m-1} & \ldots & d_{p+2m-2} \end{matrix} \right|, \qquad \left| \begin{matrix} h_p & \ldots & h_{p+m-1}\\ \ldots & \ldots & \ldots\\ h_{p+m-1} & \ldots & h_{p+2m-2} \end{matrix} \right| \tag{27} \]

are positive. Since the link terms of the continued fraction (2) are expressed through the determinants (27) ([1], p. 29, (7)), they are also positive, as was required to prove.

References

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2. S. S. Khloponin. Some transformations of continued fractions. Scientific Notes of the Mari Pedagogical Institute, vol. XXVI, pp. 445–486, Yoshkar-Ola, 1965.

3. V. E. Kornilov. Computation of elliptic integrals by means of continued fractions. Izvestiya TPI, vol. 205, 1969.

4. G. Bateman and A. Erdélyi. Higher Transcendental Functions (the hypergeometric function, Legendre functions). Moscow, Fizmatgiz, 1965.

5. V. E. Kornilov. Application of continued fractions to the computation of integrals of binomial differentials. Izvestiya TPI, vol. 131, pp. 21–25, 1965.

6. I. V. Proskuryakov. A Collection of Problems in Linear Algebra. Gostekhizdat, 1957.

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