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UDC 517.43
MATHEMATICS
S. I. OSIPOV
ON A TRANSFORMATION OF OPERATOR SUMS
(Presented by Academician A. A. Dorodnitsyn, 3 X 1966)
Let \(D = d/dt\) be the differentiation operator, \(\theta_\alpha = t^\alpha D\), \(\alpha\) any complex number,* and
\[ \theta_{\alpha_k}\theta_{\alpha_{k-1}}\cdots\theta_{\alpha_1} \tag{1} \]
denote the superposition of the operators \(\theta_{\alpha_1}, \theta_{\alpha_2}, \ldots, \theta_{\alpha_k}\) \((k = 1, 2, \ldots)\), and let \(G_n\) be the set of all functions of \(t\) differentiable \(n\) times.
The operator sum
\[ \sum_{p=1}^{n} a_{n-p}(\theta_{\alpha_k}\theta_{\alpha_{k-1}}\cdots\theta_{\alpha_1})^p \tag{2} \]
with arbitrary constant coefficients \(a_{n-p}\) can be transformed to the form
\[ \sum_{\nu=1}^{nk} Q(n,k,\nu,\alpha_i,t)D^\nu, \tag{3} \]
where \(Q(n,k,\nu,\alpha_i,t)\) are certain completely determined functions. Let
\[ Q(n,k,\nu,\alpha_i,t) = \sum_{p=\mathrm{E}\left(\frac{\nu+k-1}{k}\right)}^{n} C(k,\nu,p,\alpha_i)a_{n-p}t^{-\mu\nu}, \tag{4} \]
where
\[ \mu = k - \sum_{i=1}^{k}\alpha_i, \]
\[ C(k,\nu,p,\alpha_i) = \frac{1}{(\nu!)^2} \sum_{m=\nu}^{pk} \frac{m!}{(m-\nu)!}\,S(k,p,m,\alpha_i), \]
\[ S(k,p,m,\alpha_i) = \sum_{r=0}^{m} \frac{(-1)^{r+pk}m!}{r!(m-r)!}\, N(p,k,\alpha_i,r), \]
\[ N(p,k,\alpha_i,r) = \prod_{j=1}^{p} R(p,k,\alpha_i,r,j), \]
\[ \begin{aligned} R(p,k,\alpha_i,r,j) &= [r+(p-j+1)\mu+\alpha_k] [r+(p-j+1)\mu+\alpha_k+\alpha_{k-1}-1] \\ &\quad {}\times \cdots \times [r+(p-j+1)\mu+\alpha_k+\alpha_{k-1}+\cdots+\alpha_1-(k-1)], \end{aligned} \]
\(\mathrm{E}\left(\dfrac{\nu+k-1}{k}\right)\) is the integer part of the number \(\dfrac{\nu+k-1}{k}\).
Theorem. On the set \(G_{nk}\), the operators (2) and (3)—(4) are equivalent.
The proof is obtained by generalizing our method \((^1)\).
* The function \(t^\alpha\), as a general power function, is generally multivalued, and it is necessary to fix some single-valued branch of it.
Relying on the theorem, one can solve the following problems.
Problem 1. A linear differential operator is given,
\[ q_0(t)D^s+q_1(t)D^{s-1}+\cdots+q_{s-1}(t)D, \]
where \(q_0(t), q_1(t), \ldots, q_{s-1}(t)\) are linear combinations of generalized power functions. It is required to determine the possibility (or impossibility) of representing it in the form of a polynomial in the operator (1).
This problem is of interest in connection with methods for integrating ordinary linear differential equations supplied by the operational calculus of V. A. Ditkin \((^2)\) and its generalizations.
Problem 2. Let \(\sigma\) be any complex number,
\[ \delta(n,m)= \begin{cases} 0, & \text{if } n<m,\\ 1, & \text{if } n\geq m, \end{cases} \]
for \(p=1,2,\ldots\) and \(\xi=1,2,\ldots,(p+1)k\) the numbers \(A_{\xi,p}\) satisfy the recurrence equation
\[ A_{\xi,p+1}= \sum_{\nu=1}^{\min(\xi,pk)} \sum_{r=1}^{k} \frac{\delta(\nu+r,\xi)\,r!\,\Gamma(\nu-p\sigma+1)} {(\nu+r-\xi)!(\xi-\nu)!\,\Gamma(\xi-p\sigma-r+1)} A_{r,1}A_{\nu,p} \]
under the conditions \(A_{r,1}=c_r\), where the \(c_r\) are arbitrarily prescribed constants with the sole restriction \(c_k\ne 0\), and it is required to find \(A_{\xi,p}\) in the form of some analytic expression.
In the solution of this problem an important role is played by \(C(k,\nu,p,\alpha_i)\).
The Lah numbers \((^3)\) and the classical Stirling numbers of the second kind are special cases of \(A_{\xi,p}\).
In conclusion, let us note the formula
\[ (xy)^{-\lambda/2}e^{-y}I_\lambda(2\sqrt{xy}) = \sum_{k=0}^{n} \frac{L_k^\lambda(x)}{\Gamma(\lambda+k+1)}(-y)^k + \frac{(-y)^{n+1}}{\Gamma(\lambda+n+2)}e^{x-\xi^2}L_{n+1}^{\lambda}(\xi^2), \tag{5} \]
\[ x\geq y\geq 0,\qquad \lambda\geq -\tfrac12,\qquad \xi=\sqrt{x}+\theta\sqrt{y},\qquad |\theta|<1. \]
It can be obtained by relying on a result of B. M. Levitan \((^4)\) and on our representation \((^5)\) of the Laguerre polynomials \(L_n^\lambda(x)\) by means of the operator (1) for \(k=2\), \(\alpha_1=1+\lambda\), \(\alpha_2=-\lambda\). Formula (5) is closely related to the well-known formula of N. Ya. Sonin \((^6)\) and differs by the remainder term.
Computing Center
Academy of Sciences of the USSR
Received
27 IX 1966
REFERENCES
- S. I. Osipov, Zhurn. vychislit. matem. i matem. fiz., 4, No. 1, 149 (1964).
- V. A. Ditkin, DAN, 116, No. 1, 15 (1957).
- J. Riordan, An Introduction to Combinatorial Analysis, IL, 1963, p. 56.
- B. M. Levitan, DAN, 73, No. 2, 269 (1950).
- S. I. Osipov, Zhurn. vychislit. matem. i matem. fiz., 3, No. 1, 191 (1963).
- N. Ya. Sonin, Investigations on Cylindrical Functions and Special Polynomials, Moscow, 1954, p. 68.