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MATHEMATICS
Yu. N. FROLOV
ON THE SOLUTION OF AN EQUATION OF INFINITE ORDER IN A UNIQUENESS CLASS
(Presented by Academician I. M. Vinogradov on 3 XI 1964)
In the monograph (¹), P. Sikkema studied an operator of the form
\[ L(F) \equiv \sum_{n=0}^{\infty} c_n F^{(n)}(z), \tag{1} \]
where the function \(F(z)\) is assumed to be entire, and the sequence \(\{c_n\}\) \((n=0,1,2,\ldots)\) is a given sequence of complex numbers. The operator \(L(F)\) is called applicable to an entire function \(F(z)\) if the series (1) converges for every finite \(z\).
In (¹), necessary and sufficient conditions are given for the applicability of the operator to entire functions of the class \([\nu,\tau]\); questions of the growth of the operator \(L(F)\) depending on the growth of the function \(F(z)\) are studied; and the question of the existence of a solution of the equation \(L(F)=h(z)\), where \(h(z)\) is a given function, is considered. In the present note, operators of a more general type are considered.
Let
\[ f(z)=\sum_{n=0}^{\infty} a_n z^n,\quad a_n \ne 0\quad (n=0,1,2,\ldots) \]
be an entire function of order \(\rho\) and type \(\sigma \ne 0,\infty\), satisfying the condition: the limit exists
\[ \lim_{n\to\infty} \{n!^{1/\rho}|a_n|\}^{1/n}=(\sigma\rho)^{1/\rho}. \]
With the aid of the function \(f(z)\), generalized derivatives are generated, first introduced by A. O. Gel'fond and A. F. Leont'ev in (²). Namely, if
\[ F(z)=\sum_{n=0}^{\infty} b_n z^n, \]
then
\[ D^n F(z)=\sum_{k=0}^{\infty} b_{n+k}\frac{a_k}{a_{n+k}}z^k \]
is the generalized derivative of order \(n\) of the function \(F(z)\).
Next, let \(\{c_n\}\) \((n=0,1,2,\ldots)\) be a sequence of complex numbers. Consider the operator
\[ M(F)\equiv \sum_{n=0}^{\infty} c_n D^n F. \tag{2} \]
We note that if \(f(z)=e^z\), then the operator (2) becomes the operator (1).
The following theorem contains conditions for the applicability of the operator \(M(F)\).
Theorem 1. For the applicability of the operator \(M(F)\) to entire functions of the class \([\nu,\tau]\), it is necessary and sufficient that the function
\[ G(z)=\sum_{n=0}^{\infty} a_n^{\rho/\nu} c_n z^n \]
be an entire function of the class \([\rho,\sigma(\sigma\rho/\tau\nu)^{\rho/\nu}]\).
We note that if \(\nu=\rho\), then the condition of the theorem is equivalent to the requirement of regularity in the disk \(|z|<(\tau/\sigma)^{1/\rho}\) of the function
\[ \varphi(z)=\sum_{n=0}^{\infty} c_n z^n, \]
which is called the characteristic function for the operator \(M(F)\).
In the case when \(\nu<0\), the series \(\sum_{n=0}^{\infty} c_n z^n\) may diverge everywhere except at the point \(z=0\). Consequently, in this case the characteristic function may fail to exist altogether.
The following two theorems indicate the growth of the operator \(M(F)\) under the assumption that \(F(z)\) is an entire function of finite order and normal type.
Theorem 2. Let \(F(z)\) be an entire function of finite order \(\nu\), of normal type \(\tau\), and let the numbers \(c_0,c_1,\ldots,c_n,\ldots\) be such that the function
\[ G(z)=\sum_{n=0}^{\infty} a_n^{\rho/\nu} c_n z^n \]
is an entire function of class \(\left[\rho,\ \sigma(\sigma\rho/\tau\nu)^{\rho/\nu}\right]\).
Then the following assertions hold:
- \(h(z)=M[F(z)]\) is an entire function.
- If \(0<\nu\leqslant \rho\), then \(h(z)\in[\nu,\tau]\).
- If \(\nu>\rho\) and \(G(z)\in[(\rho,0)]\), then \(h(z)\in[(\nu,\tau)]\).
- If \(\nu>\rho\) and \(G(z)\) is an entire function of order \(\rho\) and type \(\lambda<\sigma(\sigma\rho/\tau\nu)^{\rho/\nu}\), then \(h(z)\in[\nu,d]\), where
\[ d=\tau\left\{1-\frac{(\rho\lambda)^{1/\rho}(\tau\nu)^{1/\nu}}{(\rho\sigma)^{1/\rho}(\rho\sigma)^{1/\nu}}\right\}^{\rho\nu/(\nu-\rho)}{}^{(\rho-\nu)/\rho}. \]
In the case \(0<\nu<\rho\), regarding the growth of the operator one can prove a more precise assertion.
Theorem 3. Let \(F(z)\) be an entire function of normal type \(\tau\), of finite order \(\nu\), where \(0<\nu<\rho\). Let, moreover, the numbers \(c_0,c_1,\ldots,\ldots,c_n,\ldots\) not all be zero and be such that the function
\[ G(z)=\sum_{n=0}^{\infty} a_n^{\rho/\nu} c_n z^n \]
is entire, belonging to the class \(\left[\rho,\ \sigma(\sigma\rho/\tau\nu)^{\rho/\nu}\right]\).
Then the function \(h(z)=M(F)\) is entire, of order \(\nu\) and type \(\tau\).
Consider the equation
\[ M(F)=h(z), \tag{3} \]
where \(h(z)\) is an entire function of order less than \(\rho\). For equation (3) the following holds.
Theorem 4. Let \(h(z)\) be an entire function of normal type \(\tau\), of order \(\nu\), where \(0<\nu<\rho\). Let, further, the operator \(M(F)\) be such that \(c_0\ne0\) and the function
\[ G(z)=\sum_{n=0}^{\infty} a_n^{\rho/\nu} c_n z^n \]
is entire, belonging to the class \(\left[\rho,\ \sigma(\sigma\rho/\tau\nu)^{\rho/\nu}\right]\).
Then equation (3) has one and only one solution in the class \([\nu,\tau]\), and the order and type of the solution are exactly equal, respectively, to \(\nu\) and \(\tau\).
Received
2 IX 1964
CITED LITERATURE
\(^{1}\) R. C. Sikkema, Differential Operators and Differential Equations of Infinite Order with Constant Coefficients, Groningen, 1953. \(^{2}\) A. O. Gelfond, A. F. Leont’ev, Matem. sborn., 29 (71), 3 (1951).