UDC 511.84

MATHEMATICS

I. I. BELOGRIVOV

ON THE TRANSCENDENCE AND ALGEBRAIC INDEPENDENCE OF VALUES OF CERTAIN HYPERGEOMETRIC \(E\)-FUNCTIONS

(Presented by Academician P. S. Novikov on 1 VII 1966)

In the works of A. B. Shidlovskii \((^{1,2})\) a number of general theorems were proved on the transcendence and algebraic independence of values at algebraic points of \(E\)-functions*, which are solutions of linear differential equations with polynomial coefficients. These theorems reduce the proof of algebraic independence of values of \(E\)-functions to the proof of algebraic independence of the corresponding functions over the field of rational functions.

One of the methods for proving algebraic independence of functions is a method based on the arithmetic properties of the coefficients of their power series. This method was used by K. Siegel \((^{4})\) and A. B. Shidlovskii \((^{3})\) to prove the transcendence of values of a number of concrete \(E\)-functions. In particular, A. B. Shidlovskii in \((^{3})\) applied a generalization of this method to prove the algebraic independence of the values of the functions

\[ A(z)=\sum_{n=0}^{\infty} \frac{z^{kn}}{[(\lambda+1)(\lambda+2)\ldots(\lambda+n)]^{\kappa}}, \qquad k\geq 1,\ \lambda\neq -1,-2,\ldots, \]

and of their derivatives.

In the main lemma 2 of that work the parameter \(s\) was indeterminate, enclosed within the limits from 1 to \(\nu\), and for any possible value of \(s\) one and the same prime number \(P_l\) was chosen. The assertion of this lemma is strengthened if the interval from 1 to \(\nu\) is divided into several parts and, depending on the part in which \(s\) lies, one chooses one’s own prime number. The following is true.

Lemma 1. Let \(\mu_1,\mu_2,\ldots,\mu_r\) be natural numbers, \(r\geq 1\), \(0=\mu_0<\mu_1<\cdots<\mu_r=\nu\); let \(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_r\) be real numbers, \(0\leq \varepsilon_i<1/2\), \(i=1,2,\ldots,r\);

\[ \varphi_\nu(z)=\sum_{n=0}^{\infty} a_{n,\nu}z^n, \qquad \nu=0,1,\ldots,u; \tag{1} \]

\[ f_\mu(z)=\sum_{n=}^\infty b_{n,\mu}z^n, \qquad \mu=1,2,\ldots,v. \tag{2} \]

power series with coefficients from an algebraic field \(K\), satisfying the conditions:

1. The series (1) are algebraically independent over the field of complex numbers.

2. For every natural \(t\) there exists an infinite set of systems of natural numbers \(l_1,\ldots,l_r\), \(l_i\geq t\), \(i=1,2,\ldots,r\), such that for each value of \(i\), \(1\leq i\leq r\):

a) the exact denominators of the numbers

\[ b_{l_i,\mu_{i-1}+1},\ b_{l_i,\mu_{i-1}+2},\ldots,\ b_{l_i,\mu_i} \]

* For the definition of \(E\)-functions see \((^{1,4})\).

contain the prime ideal $\mathfrak p_i$ from the field $K$ in the powers respectively
$k_{\mu_{i-1}+1}, k_{\mu_{i-1}+2}, \ldots, k_{\mu_i}$, where
$1 \leq k_{\mu_{i-1}+1}<k_{\mu_{i-1}+2}<\cdots<k_{\mu_i}$;

b) the exact denominators of the numbers

\[ a_{n,\nu},\quad n=l_i-[\varepsilon_i l_i],\ l_i-[\varepsilon_i l_i]+1,\ldots,l_i+t,\quad \nu=0,1,\ldots,u; \]

\[ b_{n,\mu},\quad n=l_i-[\varepsilon_i l_i],\ l_i-[\varepsilon_i l_i]+1,\ldots,l_i-1,\quad \mu=1,2,\ldots,v, \]

\[ b_{l_i,1},\, b_{l_i,2},\ldots,b_{l_i,\mu_i-1} \]

may contain $\mathfrak p_i$ in powers less than $k_{\mu_{i-1}+1}$;

c) the exact denominators of the numbers

\[ a_{n,\nu},\quad n=[l_i/2],\ [l_i/2]+1,\ldots,l_i-[\varepsilon_i l_i]-1,\quad \nu=0,1,\ldots,u; \]

\[ b_{n,\mu},\quad n=[l_i/2],\ [l_i/2]+1,\ldots,l_i-[\varepsilon_i l_i]-1,\quad \mu=1,2,\ldots,v, \]

may contain $\mathfrak p_i$ in powers less than $\frac12 k_{\mu_{i-1}+1}$;

d) the exact denominators of the numbers

\[ a_{n,\nu},\quad n=[l_i/2],\ [l_i/2]+1,\ldots,l_i-[\varepsilon_i l_i]-1,\quad \nu=0,1,\ldots,u; \]

\[ b_{n,\mu},\quad n=[l_i/2],\ [l_i/2]+1,\ldots,l_i-[\varepsilon_i l_i]-1,\quad \mu=1,2,\ldots,v, \]

do not contain $\mathfrak p_i$.

Then the $u+v+1$ power series (1) and (2) are algebraically independent over the field of complex numbers.

Using the fundamental theorem of A. B. Shidlovskii (¹), Lemma 1, and several auxiliary propositions on the distribution of prime numbers in various arithmetic progressions, one obtains a series of theorems formulated below on the transcendence and algebraic independence of values at algebraic points of certain hypergeometric $E$-functions that are solutions of linear differential equations of arbitrary orders with polynomial coefficients.

Let $\lambda,\lambda_1,\ldots,\lambda_s$ be complex numbers; let $m_1,m_2,\ldots,m_s$ be nonnegative integral rational numbers; $m=m_1+\cdots+m_s \geq 1$. Denote $[\lambda,0]=1$, $[\lambda,n]=\lambda(\lambda+1)\cdots(\lambda+n-1)$, $n\geq1$, and

\[ A_{m,s}(z)=\sum_{n=0}^{\infty} \frac{1}{[\lambda_1+1,n]^{m_1}[\lambda_2+1,n]^{m_2}\cdots[\lambda_s+1,n]^{m_s}} \left(\frac{z}{m}\right)^{mn}, \tag{3} \]

\[ \lambda_1,\ldots,\lambda_s\ne 1,-2,\ldots \]

Further, let $t_i=m_i$, $i=1,2,\ldots,s-1$; let $t_s$ be a natural number, $t_s\geq m_s$; $l=m_1+\cdots+m_{s-1}+t_s$; $m_0=q_0=0$; $q_i=t_1+\cdots+t_i$, $i=1,2,\ldots,s$;

\[ A_{m,s,\mu}(z)=1+ \tag{4} \]

\[ +\sum_{n=1}^{\infty} \frac{(z/m)^{mn}} {[\lambda_1+1,n]^{m_1}\cdots[\lambda_{i-1}+1,n]^{m_{i-1}} [\lambda_i+1,n-1]^{m_i}\cdots[\lambda_s+1,n-1]^{m_s} (\lambda_i+n)^{\mu-q_{i-1}}} \]

\[ \lambda_i\ne -1,-2,\ldots;\quad \mu=q_{i-1}+1,q_{i-1}+2,\ldots,q_i,\quad i=1,2,\ldots,s. \]

In particular, for $s=2$ the functions (4) have the form

\[ A_{m,2,\mu}(z)=1+\sum_{n=1}^{\infty} \frac{1}{[\lambda_1+1,n-1]^{m_1}[\lambda_2+1,n-1]^{m_2}(\lambda_1+n)^\mu} \left(\frac{z}{m}\right)^{mn}, \]

\[ \mu=1,2,\ldots,q_1, \]

\[ A_{m,2,\mu}(z)=1+\sum_{n=1}^{\infty} \frac{1}{[\lambda_1+1,n]^{m_1}[\lambda_2+1,n-1]^{m_2}(\lambda_2+n)^{\mu-q_1}} \left(\frac{z}{m}\right)^{mn}, \]

\[ \mu=q_1+1,q_1+2,\ldots,l. \]

Theorem 1. Let \(\lambda_1,\lambda_2\) be rational numbers, \(\lambda_1,\lambda_2\ne -1,-2,\ldots\), and, if one of them is an integer, let the other be different from half an odd number, and let \(\alpha\ne 0\) be any algebraic number.

Then: 1) the \(l\) numbers \(A_{m,2,\mu}(\alpha)\), \(\mu=1,2,\ldots,l\), are algebraically independent; 2) the \(m\) numbers \(A_{m,2}(\alpha), A'_{m,3}(\alpha),\ldots,A^{(m-1)}_{m,2}(\alpha)\) are algebraically independent.

From Theorem 1, for \(\lambda_1=0,\lambda_2=\lambda, m_1=m_2=1\), there follows the known result of C. L. Siegel \((^4)\) on the algebraic independence of the values of the functions \(K_\lambda(z), K'_\lambda(z)\).

Theorem 1 also generalizes Theorem 1 of A. B. Shidlovskii from the paper \((^5)\) and refines the result of the latter. In Theorem 1 of the paper \((^5)\), which is obtained from Theorem 1 for \(m_1=m_2=1\) and \(\lambda_1=\mu, \lambda_2=\lambda\), a restriction is imposed on the values of the parameters \(\lambda,\mu\), namely that the difference \(\mu-\lambda\) be different from half an odd number. In our theorem this restriction is retained only in the case when one of the values \(\lambda\) or \(\mu\) is an integer.

Theorem 1 contains Theorem 6 and, in part, Theorems 7, 8 of the paper \((^3)\).

Theorem 2. Let \(\lambda_1,\lambda_2,\lambda_3\) be rational numbers, \(\lambda_1,\lambda_2,\lambda_3\ne -1,-2,\ldots\), let \(\lambda_1+\lambda_2\) be different from an integer, and let \(\lambda_3\) and at least one of the numbers \(2\lambda_1-\lambda_2\) and \(2\lambda_2-\lambda_1\) be integers; moreover, if one of the numbers \(\lambda_1\) and \(\lambda_2\) is equal to \((2k_1-1)/2\), then the other is not equal to \(k_2/4\), where \(k_1,k_2\) are integers, and let \(\alpha\ne 0\) be any algebraic number.

Then: 1) the \(l\) numbers \(A_{m,3,\mu}(\alpha)\), \(\mu=1,2,\ldots,l\), are algebraically independent; 2) the \(m\) numbers \(A_{m,3}(\alpha), A'_{m,3}(\alpha),\ldots,A^{(m-1)}_{m,3}(\alpha)\) are algebraically independent.

In the special case when \(m_1=m_2=m_3=1\), \(\lambda_1=\lambda,\lambda_2=\mu,\lambda_3=0\), the function \(A_{3,3}(z)\) becomes the function \(K_{\lambda,\mu}(z)\), considered by V. A. Oleinikov in the paper \((^6)\), where it is shown that for any rational values \(\lambda_1,\lambda_2\) for which none of the numbers \(|2\lambda_1-\lambda_2|, |2\lambda_2-\lambda_1|, |\lambda_2-\lambda_1|\) is natural, the three numbers \(A_{3,3}(\alpha), A'_{3,3}(\alpha), A''_{3,3}(\alpha)\) are algebraically independent for any algebraic value \(\alpha\ne 0\).

Theorem 2 gives a certain refinement of this result. The numbers \(A_{3,3}(\alpha), A'_{3,3}(\alpha), A''_{3,3}(\alpha)\) will also be algebraically independent when \(2\lambda_1-\lambda_2\) or \(2\lambda_2-\lambda_1\) is an integer, while \(\lambda_1+\lambda_2\) is different from an integer, but under the condition that if one of the numbers \(\lambda_1\) or \(\lambda_2\) is equal to \((2k_1-1)/2\), then the other is not equal to \(k_2/4\), where \(k_1,k_2\) are integers.

Theorem 3. Let \(\lambda_i=b_i/a_i\), \(\lambda_i\ne -1,-2,\ldots\), where \(a_i,b_i\) are integers, \(a_i\ge 1\), \((a_i,b_i)=1\), \(i=1,2,\ldots,s\), \(s\ge 2\), \(a_i a_j\ne 2\), \((a_i,a_j)=1\), \(1\le i<j\le s\), and let \(\alpha\ne 0\) be any algebraic number.

Then: 1) the \(l\) numbers \(A_{m,s,\mu}(\alpha)\), \(\mu=1,2,\ldots,l\), are algebraically independent; 2) the \(m\) numbers \(A_{m,s}(\alpha), A'_{m,s},\ldots,A^{(m-1)}_{m,s}(\alpha)\) are algebraically independent.

Theorem 4. Let \(\lambda_1,\ldots,\lambda_s\) be rational numbers, \(\lambda_1,\ldots,\lambda_s\ne -1,-2,\ldots\), whose fractional parts satisfy the conditions

\[ \{\lambda_1\}<\{\lambda_2\}<\cdots<\{\lambda_s\},\qquad 2\{\lambda_1\}>\{\lambda_{s-1}\}, \]

and let \(\alpha\ne 0\) be any algebraic number.

Then: 1) the \(l\) numbers \(A_{m,s,\mu}(\alpha)\), \(\mu=1,2,\ldots,l\), are algebraically independent; 2) the \(m\) numbers \(A_{m,s}(\alpha), A'_{m,s}(\alpha),\ldots,A^{(m-1)}_{m,s}(\alpha)\) are algebraically independent.

Put in expressions (3) and (4) \(s=p+h\), where \(p,h\) are nonnegative integers.

Theorem 5. Let \(\lambda_i=b_i/a_i\), \(\lambda_i\ne -1,-2,\ldots\), where \(a_i,b_i\) are integers, \(a_i\ge 1\), \((a_i,b_i)=1\), \(i=1,2,\ldots,s\), \(s\ge 2\); let \(A'\) and \(A''\) be the least common multiples of the numbers, respectively, \(a_1,a_2,\ldots,a_s\) and \(a_{p+1},a_{p+2},\ldots,a_s\); let \((A',A'')=d\), \(1\le d\le 2\), let the fractional parts \(\{\lambda_i\}>1/2\) for \(i\le p\), \(\{\lambda_i\}<1/2\) for \(i>p\), and let \(\alpha\ne 0\) be any algebraic number.

Then: 1) the \(l\) numbers \(A_{m,s,\mu}(\alpha)\), \(\mu=1,2,\ldots,l\), are algebraically independent; 2) the \(m\) numbers \(A_{m,s}(\alpha), A'_{m,s}(\alpha),\ldots,A^{(m-1)}_{m,s}(\alpha)\) are algebraically independent.

In particular, for \(h=0\) or \(p=0\), Theorem 5 contains

Theorem 6. Let \(\lambda_1,\ldots,\lambda_s\) be rational numbers whose fractional parts, different from zero, are either greater than one half or less than one half, and let \(\alpha\ne 0\) be any algebraic number.

Then: 1) the \(l\) numbers \(A_{m,s,\mu}(\alpha)\), \(\mu=1,2,\ldots,l\), are algebraically independent; 2) the \(m\) numbers \(A_{m,s}(\alpha), A'_{m,s}(\alpha),\ldots,A^{(m-1)}_{m,s}(\alpha)\) are algebraically independent.

The author expresses his deep gratitude to Prof. A. B. Shidlovskii for posing the problem and for his help.

State Pedagogical Institute
named after V. I. Lenin

Received
23 VI 1966

REFERENCES

1. A. B. Shidlovskii, Izv. AN SSSR, Ser. Mat., 23, No. 1, 35 (1959).
2. A. B. Shidlovskii, Izv. AN SSSR, Ser. Mat., 26, No. 6, 877 (1962).
3. A. B. Shidlovskii, Vestn. Mosk. Univ., No. 5, 44 (1961).
4. C. Siegel, Abh. Preuss. Acad. Wiss., No. 1, 1 (1929–1930).
5. A. B. Shidlovskii, DAN, 96, No. 4, 697 (1954); Uch. zap. MGU, vol. 186, Mathematics, 9, 11 (1958).
6. V. A. Oleinikov, DAN, 166, No. 3, 540 (1966).