MATHEMATICS

M. D. KHRIPTUN

ON A CLASS OF ENTIRE FUNCTIONS

(Presented by Academician V. I. Smirnov, 1 XI 1956)

It can be shown \((^{3,4})\) that the function

\[ U_0^{(1)}\left(3\sqrt[3]{s}\right)=\sum_{n=0}^{\infty}\frac{s^n}{(2n)!\,n!} \tag{1} \]

in the Cauchy problem for an ordinary linear differential operator \(L\) of the 2nd order plays the same role as, for self-adjoint operators, is played by the function \(e^s\), and for differential operators of the 1st order by the Bessel function
\[ I_0\left(2\sqrt{s}\right)=\sum_{n=0}^{\infty}\frac{s^n}{n!\,n!}, \]
namely: for \(s=zu^2\) the function (1) will be the kernel of the integral representation of the operator \(e^{zA}\), where \(A=L_0^{-1}\) is the operator inverse to \(L\) for zero initial data (at the point \(x=0\)).

Making in (1) the substitution \(s=(z/3)^3\) (\(z\) a complex variable), replacing \(n!\) by \(\Gamma(n+p+1)\), and multiplying by \((z/3)^p\), we obtain the function

\[ U_p^{(1)}(z)=\sum_{n=0}^{\infty}\frac{(z/3)^{p+3n}}{(2n)!\,\Gamma(n+p+1)}, \tag{2} \]

which satisfies the equation

\[ z^3U'''(z)+{}^{3}/_{2}\,z^2U''(z)-{}^{1}/_{2}\,z(6p^2+3p+1)U'(z)+ \]
\[ +(2p^3+3p^2-{}^{1}/_{4}z^3)U(z)=0. \tag{3} \]

It turned out that its solutions possess properties very similar to the properties of cylindrical functions. The purpose of the present note is to give a brief account of these properties.

§ 1. The given equation has two singular points: a regular one \(z=0\), and an irregular one \(z=\infty\) \((^{1})\), p. 491).

Applying the usual arguments, we obtain:

I. If \(p\ne k/3,\ p\ne(2k-3)/6\), where \(k=0,\ \pm1,\ \pm2,\ldots\), then three independent solutions of equation (3) are as follows: the first has the form given by formula (2), while the other two take the form

\[ U_{p+{}^{1}/_{2}}^{(2)}(z)=\sum_{n=0}^{\infty}\frac{(z/3)^{p+{}^{3}/_{2}+3n}}{(2n+1)!\,\Gamma(n+p+{}^{3}/_{2})}, \tag{4} \]

\[ U_{-2p}^{(3)}(z)=\sum_{n=0}^{\infty}\frac{\Gamma(2p-2n)(z/3)^{-2p+3n}}{\Gamma(2p)\,n!}. \tag{5} \]

We shall call these solutions, respectively, a solution of the first kind of the first type with index \(p\); of the first kind of the second type with index \(p+\frac{3}{2}\); of the first kind of the third type with index \(-2p\).

II. Let \(p=k/3\).

a) \(k=0,1,2,\ldots\). The solutions corresponding to the indices \(p\) and \(p+\frac{3}{2}\) remain, in form, the same as (2), (4). The solution corresponding to the index \(-2p\) is constructed by the general Frobenius method \(((^{1}),\) pp. 534—545):

\[ M_{-2p}^{(1)}(z)= \sum_{n=0}^{\infty} \frac{1}{(2n)!\,\Gamma(n+p+1)} \{3\ln z-\psi(n)-\psi(p+n)-\psi(n-\tfrac{1}{2})\} \left(\frac{z}{3}\right)^{p+3n} + \]

\[ + \sum_{n=0}^{p-1} \frac{\Gamma(2p-2n)}{\Gamma(2p)n!} \left(\frac{z}{3}\right)^{-2p+3n}, \tag{6} \]

where

\[ \psi(\tau)=\left[\frac{d}{dt}\log\Gamma(t+1)\right]_{t=\tau}. \]

We shall call this solution a solution of the second kind of the first type with index \(-2p\).

b) If \(k=-1,-2,-3,\ldots\), then the form of the solutions corresponding to the indices \(-2p\) and \(p+\frac{3}{2}\) remains the same as in case I, while the solution corresponding to the index \(p\) is constructed in the same way as \(M_{-2p}^{(1)}(z)\); it has the form:

\[ M_{p}^{(3)}(z)= \sum_{n=0}^{\infty} \frac{\Gamma(2p-2n)}{\Gamma(2p)n!} \{3\ln z-\psi(n)-\psi(-p+n)-\psi(-p-\tfrac{1}{2}+n)\} \left(\frac{z}{3}\right)^{-2p+3n} + \]

\[ + \sum_{n=0}^{-p-1} \frac{1}{(2n)!\,\Gamma(n+p+1)} \left(\frac{z}{3}\right)^{p+3n}; \tag{7} \]

we shall call it a solution of the second kind of the third type with index \(p\).

III. Let \(p=(2k-3)/6\).

a) \(k=2,3,4,\ldots\). The solutions corresponding to the indices \(p+\frac{3}{2}\) and \(p\) have the form (4) and (2), while the solution corresponding to the index \(-2p\) takes the form

\[ M_{-2p}^{(2)}(z)= \sum_{n=0}^{\infty} \frac{1}{(2n+1)!\,\Gamma(n+p+\tfrac{3}{2})} \{3\ln z-\psi(n)-\psi(n+\tfrac{1}{2})- \]

\[ -\psi(p+n+\tfrac{1}{2})\} \left(\frac{z}{3}\right)^{p+\frac{3}{2}+3n} + \sum_{n=0}^{p+\frac{1}{2}-1} \frac{\Gamma(2p-2n)}{\Gamma(2p)n!} \left(\frac{z}{3}\right)^{-2p+3n}; \tag{8} \]

we call it a solution of the second kind of the second type with index \(-2p\).

b) \(k=0,\pm1,-2,-3,\ldots\). The solution corresponding to the index \(p+\frac{3}{2}\) has the form

\[ M_{p+\frac{3}{2}}^{(3)}(z)= \]

\[ = \sum_{n=0}^{\infty} \frac{\Gamma(2p-2n)}{\Gamma(2p)n!} \{3\ln z-\psi(n)-\psi(-p+n)-\psi(-p-\tfrac{1}{2}+n)\} \left(\frac{z}{3}\right)^{-2p+3n} + \]

\[ + \sum_{n=0}^{-p-\frac{1}{2}-1} \frac{1}{(2n+1)!\,\Gamma(n+p+\tfrac{3}{2})} \left(\frac{z}{3}\right)^{p+\frac{3}{2}+3n}, \tag{9} \]

we call it a solution of the second kind of the third type with index \(p+\frac{3}{2}\), and the remaining two solutions have the form (2) and (5), respectively.

IV. Let us single out the case \(p=0\); then all three independent solutions take the form:

\[ U_{0}^{(1)}(z)= \sum_{n=0}^{\infty} \frac{(z/3)^{3n}}{(2n)!n!}, \]

\[ M_0^{(1)}(z)=\sum_{n=0}^{\infty}\frac{1}{(2n)!\,n!}\left\{\ln z-\frac{2}{3}\psi(n)-\frac{1}{3}\psi\left(n-\frac{1}{2}\right)\right\}\left(\frac{z}{3}\right)^{3n}, \]

\[ U_{1/2}^{(2)}(z)=\sum_{n=0}^{\infty}\frac{(z/3)^{1/2+3n}}{(2n+1)!\,\Gamma(n+3/2)}. \]

In all these formulas
\[ \psi(r)=\left[\frac{d}{dt}\log\Gamma(t+1)\right]_{t=r}. \]
Solutions of the first and second kinds of equation (3) are analogous to Bessel functions of the first and second kinds.

§ 2. The generating function and integral representation for integer indices. It is easy to verify the relation:
\[ e^{zt/3}\operatorname{ch}\frac{z}{3\sqrt{t}} = \sum_{p=-\infty}^{\infty} t^p U_p^{(1)}(z), \tag{10} \]
where in formula (2) the terms corresponding to the poles of the function \(\Gamma(n+p+1)\) are absent. Hence
\[ U_p^{(1)}(z)=\frac{1}{2\pi i}\oint_{(0)} t^{-p-1}e^{zt/3}\operatorname{ch}\frac{z}{3\sqrt{t}}\,dt \quad (p=0,\pm1,\pm2,\ldots). \tag{11} \]

Similarly
\[ \sqrt{t}\,e^{zt/3}\operatorname{sh}\frac{z}{3\sqrt{t}} = \sum_{p=-\infty}^{\infty} t^p U_{p+1/2}^{(2)}(z). \tag{12} \]

Hence
\[ U_{p+1}^{(2)}(z)=\frac{1}{2\pi i}\oint_{(0)} t^{-p-1}\sqrt{t}\,e^{zt/3}\operatorname{sh}\frac{z}{3\sqrt{t}}\,dt. \tag{13} \]

§ 3. Integral representations of the function \(U\) with an arbitrary index. Substituting the known expression
\[ \frac{1}{\Gamma(n+p+1)}=\frac{1}{2\pi i}\int_{l'} e^\tau \tau^{-(n+p+1)}\,d\tau \tag{14} \]
((2), p. 275) into (2), after transformations we obtain
\[ U_p^{(1)}(z)=\frac{1}{2\pi i}\int_{l'}\left(\frac{z}{3}\right)^p \tau^{-p-1}e^\tau \operatorname{ch}\left(\frac{z}{3}\sqrt{\frac{z}{3\tau}}\right)\,d\tau. \]

Assuming that
\[ |\arg z|<\pi/2 \tag{15} \]
and putting \(\tau=zt/3\), we obtain
\[ U_p^{(1)}(z)=\frac{1}{2\pi i}\int_l t^{-p-1}e^{zt/3}\operatorname{ch}\frac{z}{3\sqrt{t}}\,dt, \tag{16} \]
where as the path of integration one may take the former contour \(l'\) (on the basis of condition (15)).

In the same way we obtain two other representations:
\[ U_{p+1/2}^{(2)}(z)=\frac{1}{2\pi i}\int_l t^{-p-1}e^{zt/3}\operatorname{sh}\frac{z}{3\sqrt{t}}\,dt, \tag{17} \]
\[ U_{-2p}^{(3)}(z)=\frac{1}{2\pi i}\int_l t^{2p-1}e^{\frac{z}{3}\left(\frac{1}{t^2}+t\right)}\,dt. \tag{18} \]

Introducing \(X_1(z)=U_p^{(1)}(z)+U_{p+1/2}^{(2)}(z)\) and \(X_2(z)=U_p^{(1)}(z)-U_{p+1/2}^{(2)}(z)\), putting \(u=1/\sqrt{t}\) for \(X_2(z)\), \(u=-1/\sqrt{t}\) for \(X_2(z)\), we bring their representa-

…to the form (18); now replacing \(u=e^{i\omega}\) and returning to the functions \(U\), we obtain the representations

\[ U_p^{(1)}(z)=\frac{1}{4\pi}\int_C \exp\left[\frac{z}{3}\left(e^{-2i\omega}+e^{i\omega}\right)+2pi\omega\right]\,d\omega, \tag{19} \]

\[ U_{p+\frac12}^{(2)}(z)=\frac{1}{4\pi}\int_{C'} \exp\left[\frac{z}{3}\left(e^{-2i\omega}+e^{i\omega}\right)+2pi\omega\right]\,d\omega, \tag{20} \]

\[ U_{-2p}^{(3)}(z)=\frac{1}{2\pi}\int_{C''} \exp\left[\frac{z}{3}\left(e^{-2i\omega}+e^{i\omega}\right)+2pi\omega\right]\,d\omega \tag{21} \]

through one and the same function over different contours, shown respectively in Fig. 1 a, b, and c.

Fig. 1

§ 4. Addition theorems. If

\[ R=\sqrt[3]{(r_1 e^{-i\theta/2}-r_2)^2(r_1 e^{i\theta}-r_2)}, \]

where \(r_1,r_2\) are complex variables, then

\[ U_0^{(1)}(\lambda R)= \sum_{m=-\infty}^{\infty} U_m^{(1)}(\lambda r_1)U_{-m}^{(1)}(-\lambda r_2)e^{im\theta} + \]

\[ + \sum_{m=-\infty}^{\infty} U_{m+1}^{(2)}(\lambda r_1)U_{-m+2}^{(2)}(-\lambda r_2)e^{i\theta(m-\frac12)}. \tag{22} \]

For \(\theta=0\) a simpler formula is obtained:

\[ U_0^{(1)}(\lambda(r_1-r_2))= \sum_{m=-\infty}^{\infty} U_m^{(1)}(\lambda r_1)U_{-m}^{(1)}(-\lambda r_2) + \]

\[ + \sum_{m=-\infty}^{\infty} U_{m+1}^{(2)}(\lambda r_1)U_{-m+2}^{(2)}(-\lambda r_2). \tag{23} \]

Formula (22) admits a generalization to the case of an arbitrary index \(\nu\):

\[ U_\nu^{(1)}(\lambda R)e^{i\nu\psi}= \sum_{m=-\infty}^{\infty} U_m^{(1)}(\lambda r_1)U_{\nu-m}^{(1)}(-\lambda r_2)e^{im\theta} + \]

\[ + \sum_{m=-\infty}^{\infty} U_{m+1}^{(2)}(\lambda r_1)U_{\nu-m+2}^{(2)}(-\lambda r_2)e^{i\theta(m-\frac12)}, \tag{24} \]

where \(\psi\) must satisfy the relations: \(R\cos\psi=r_1\cos\theta-r_2\); \(R\sin\psi=r_1\sin\theta\), or \(e^{i\psi}=(r_1e^{i\theta}-r_2)/R\).

§ 5. There also hold differential recurrence relations.

In conclusion I express my sincere gratitude to the supervisor of the present work, M. K. Fage, for valuable instructions and assistance rendered to me in the course of carrying out the work.

Chernivtsi
State University

Received
29 X 1956

References Cited

1. E. L. Ince, Ordinary Differential Equations, Kharkov, 1939.
2. V. I. Smirnov, A Course of Higher Mathematics, 3, part 2, Moscow, 1953.
3. M. K. Fage, DAN, 95, No. 4, 721 (1954).
4. M. K. Fage, DAN, 99, No. 6, 909 (1954).