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MATHEMATICS
Yu. L. RABINOVICH and S. V. NESTEROV
GENERAL FORM OF LINEAR DIFFERENTIAL EQUATIONS WHOSE ORDER IS REDUCED BY MEANS OF THE OPERATOR OF GENERALIZED DIFFERENTIATION \(D^\alpha\)
(Presented by Academician I. G. Petrovskii, December 2, 1960)
1. Introduction. The operator \(D^\alpha\), i.e. the integral transformation with the Euler–Cauchy kernel \((z-\zeta)^{-\alpha-1}\) \((^{1})\), transforms a differential expression of order \(n\) of the form
\[ \mathcal{L}[u]=\sum_{k=0}^{n} p_k(z)\frac{d^{\,n-k}u}{dz^{\,n-k}}, \tag{1} \]
where \(p_k(z)\) are polynomials of degrees \(m_k\), into the expression
\[ T_\alpha[v]=\sum_{k=0}^{l} q_k(z,\alpha)\frac{d^{\,l-k}v}{dz^{\,l-k}}, \tag{2} \]
where
\[ l=\max(m_0,\; m_1+1,\; m_2+2,\ldots,\; m_n+n); \tag{3} \]
\[ q_k(z,\alpha)=\sum_{h=0}^{\min(k,n)} \binom{\alpha}{k-h}\,p_h^{(k-h)}(z). \tag{4} \]
Between the expressions \(\mathcal{L}\) and \(T_\alpha\) there is the relation
\[ D^\alpha \mathcal{L}[u]=T_\alpha[D^{\alpha+n-l}u]\quad (^{2,3}). \tag{5} \]
Let
\[ m_0=n;\qquad m_k\leq n-k\quad \text{for } k=1,2,\ldots,n. \tag{6} \]
Then
\[ l=n. \tag{7} \]
Putting
\[ p_{n-k}(z)=\sum_{h=0}^{k} p_{n-k,h}z^{k-h},\qquad q_{n-k}(z,\alpha)=\sum_{h=0}^{k}\Phi_{kh}(\alpha)z^h, \tag{8} \]
we find
\[ \Phi_{k0}(\alpha)= \sum_{h_1=0}^{n-k} p_{n-k-h_1,k}\,\alpha(\alpha-1)\cdots(\alpha-h_1+1); \tag{9} \]
\[ \Phi_{kh}(\alpha)=\frac{1}{h!}\Delta^h\Phi_{k-h,0}(\alpha), \tag{10} \]
where \(\Delta^h\) is the finite difference of order \(h\).
Under conditions (6), \(z=\infty\) is a regular singular point of the equation
\[ \mathscr{L}[u]=0 \tag{11} \]
with characteristic exponents \(\rho_1,\rho_2,\ldots,\rho_n\), satisfying the determining equation
\[ \Phi_{00}(\rho)=0. \tag{12} \]
In the absence of logarithms, to the exponent \(\rho_k\) there corresponds a canonical solution \(u_k(z)\) of equation (11) at \(z=\infty\), expanded in a neighborhood of \(z=\infty\) in the series
\[ u_k(z)=\sum_{m=0}^{\infty} A_{km} z^{\rho_k-m}. \tag{13} \]
The operator \(D_{\infty z}^{\alpha}\) is defined by the contour integral
\[ D_{\infty z}^{\alpha} f(z)= \frac{\Gamma(\alpha+1)} {2\pi i\left(1-e^{2(\alpha-\beta)\pi i}\right)} \int_C (\zeta-z)^{-\alpha-1} f(\zeta)\,d\zeta, \tag{14} \]
where \(C=\zeta_0 z^+ \zeta_0 \infty^+ \zeta_0 z^- \zeta_0 \infty^- \zeta_0\), with \(\zeta_0\) an arbitrary regular point of the integrand, the loops \(\zeta_0\infty^\pm \zeta_0\) enclosing all singular points, and \(\zeta_0 z^\pm \zeta_0\) only the singular point \(\zeta=z\) of the integrand\({}^{4}\). In this case
\[ D_{\infty z}^{\alpha} z^\beta = e^{\pm \pi \alpha i} \frac{\Gamma(\alpha-\beta)}{\Gamma(-\beta)} z^{\beta-\alpha}. \tag{15} \]
To the canonical solutions (13) of equation (11) there correspond canonical solutions of the equation \(T_\alpha[v]=0\) of the form
\[ v_k(z,\alpha)=D_{\infty z}^{\alpha}u_k(z) = e^{\pm \pi \alpha i} \sum_{m=0}^{\infty} \frac{\Gamma(\alpha-\rho_k+m)}{\Gamma(m-\rho_k)} A_{km} z^{\rho_k-\alpha-m}. \tag{16} \]
If \(\alpha=\rho_k-h\), where \(h=0,1,2,\ldots\), then \(v_k\equiv\infty\), and
\[ {}^{*}v_k(z)= \lim_{\alpha\to \rho_k-h} \frac{v_k(z,\alpha)}{\Gamma(\alpha-\rho_k)} = \]
\[ = e^{\pm \pi(\rho_k-h)i} \sum_{m=0}^{h} (-1)^m \frac{h(h-1)\ldots(h-m+1)}{\Gamma(m-\rho_k)} A_{km} z^{h-m}. \tag{17} \]
- In the present paper we investigate the properties of those expressions \(\mathscr{L}[u]\) whose order is lowered by means of the operator \(D^\alpha\). If, under conditions (6), the relations
\[ q_n(z,\alpha)\equiv 0,\qquad q_{n-1}(z,\alpha)\equiv 0,\ldots,\qquad q_{n-\mu+1}(z,\alpha)\equiv 0 \tag{18} \]
hold for any \(z\) and fixed \(\alpha=\alpha_1\), then, putting
\[ \Lambda[w]= \sum_{k=0}^{n-\mu} q_k(z,\alpha_1)\, \frac{d^{\,n-\mu-k}w}{dz^{\,n-\mu-k}}, \tag{19} \]
we obtain
\[ D^{\alpha_1}\mathscr{L}[u]=\Lambda[D^{\alpha_1+\mu}u]. \tag{20} \]
In this case the operator \(D^{\alpha_1}\) transforms the expression of \(n\)-th order \(\mathscr{L}[u]\) into the expression of \((n-\mu)\)-th order \(\Lambda[w]\), where \(w=D^{\alpha_1+\mu}u\).
Theorem. Let there be given natural numbers \(n\) and \(\mu<n\), polynomials \(p_0(z),p_1(z),\ldots,p_{n-\mu}(z)\) of degrees \(n,n-1,\ldots,\mu\), and a complex number \(\omega_1\)
Then equations (18) for \(a=a_1\) uniquely determine the polynomials \(p_{n-\mu+1}(z),\ldots,p_n(z)\). The corresponding expression \(\mathcal L[u]\) satisfies relation (20), i.e., the operator \(D^{\alpha_1}\) lowers the order of \(\mathcal L[u]\) by \(\mu\) units. Equations (18) for \(a=a_1\) are equivalent to the equations
\[ \Phi_{00}(\alpha_1)=0;\qquad \Phi_{00}(\alpha_1+1)=0;\ \ldots;\ \Phi_{00}(\alpha_1+\mu-2)=0; \tag{21} \]
\[ \Phi_{00}(\alpha_1+\mu-1)=0; \]
\[ \Phi_{10}(\alpha_1)=0;\qquad \Phi_{10}(\alpha_1+1)=0;\ \ldots;\ \Phi_{10}(\alpha_1+\mu-2)=0; \]
\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]
\[ \Phi_{\mu-2,0}(\alpha_1)=0;\qquad \Phi_{\mu-2,0}(\alpha_1+1)=0; \tag{22} \]
\[ \Phi_{\mu-1,0}(\alpha_1)=0. \]
Equations (21) mean that the order \(\alpha_1\) of the operator \(D^{\alpha_1}\), as well as the numbers \(\alpha_1+1,\ldots,\alpha_1+\mu-1\), are characteristic exponents of equation (11) at \(z=\infty\). Suppose that, for any natural number \(h\),
\[ \Phi_{00}(\alpha_1-h)\ne 0. \tag{23} \]
Then from conditions (22) it follows that to the exponents \(\rho_k=\alpha_1+k-1\) \((k=1,2,\ldots,\mu)\) there belong \(\mu\) linearly independent logarithm-free solutions \(u_k(z)\) of equation (11) of the form (13).
- Since for \(1\le k\le \mu\), \(\rho_k-\alpha_1=k-1\), the solutions \(v_1(z,\alpha_1),\ldots,v_\mu(z,\alpha_1)\) of the equation \(T_{\alpha_1}[v]=0\) lose their meaning and must be replaced by the solutions
\[ v_k^*(z)=\lim_{\alpha\to\alpha_1} \frac{v_k(z,\alpha)}{\Gamma(\alpha-\alpha_1-k+1)}, \]
where \(v_k^*(z)\) is a polynomial of degree \(k-1\). If the differences \(\rho_{\mu+1}-\alpha_1,\ldots,\rho_n-\alpha_1\) are not integers, then the functions
\[ w_{\mu+1}=D_{\infty z}^{\alpha_1+\mu}u_{\mu+1}(z),\ldots, w_n(z)=D_{\infty z}^{\alpha_1+\mu}u_n(z) \tag{24} \]
form a fundamental system of the equation
\[ \Lambda[w]=0. \tag{25} \]
On the other hand, for \(1\le k\le\mu\),
\[ \lim_{\alpha\to\alpha_1} \sum_{j=0}^{\mu-1} q_{n-j}(z,\alpha)\, \frac{d^j v_k(z,\alpha)}{dz^j} = Q_k(z), \]
where \(Q_k(z)\) is a polynomial of degree \(k-1\), so that the functions \(w_k=D_{\infty z}^{\alpha_1+\mu}u_k\) for \(k\le\mu\) satisfy the equations \(\Lambda[w_k]=-Q_k\), whence it follows that all \(n\) functions \(w_i=D_{\infty z}^{\alpha_1+\mu}u_i(z)\) \((i=1,2,\ldots,n)\) form a fundamental system of the equation
\[ \frac{d^\mu}{dz^\mu}\Lambda[w]=0. \tag{26} \]
The fundamental system of the equation
\[ D^\alpha\mathcal L[u]=0 \tag{27} \]
consists of the functions
\[ 1,\ z,\ z^2,\ldots,z^{\mu-1},\ D_{\infty z}^{\alpha_1}u_{\mu+1}(z),\ldots,D_{\infty z}^{\alpha_1}u_n(z). \]
Moscow State University
named after M. V. Lomonosov
Received
16 XI 1960
References Cited
- E. L. Ince, Ordinary Differential Equations, Ch. VIII, 1939.
- N. Ya. Sonin, Matem. sborn., 6, 1 (1872).
- P. A. Nekrasov, Matem. sborn., 14 (1888).
- R. Courant, D. Hilbert, Methods of Mathematical Physics, 2, 1951, pp. 520—529.