V. M. ARUTIUNIAN, R. M. MURADIAN, and A. A. SOKOLOV

ASYMPTOTIC EXPRESSION FOR THE CONFLUENT HYPERGEOMETRIC FUNCTION

(Presented by Academician N. N. Bogoliubov, June 2, 1958)

The asymptotic behavior of solutions of a differential equation of the form

\[ u''+f(x)u=0, \tag{1} \]

where \(f(x)\) is a function of one or several parameters, can be studied by constructing a so-called “nearby equation,” on the basis that differential equations which are approximately the same for all \(x\) must have approximately the same solutions. We shall give a method for constructing such an equation; in particular, we shall apply the results obtained to finding asymptotic formulas for Whittaker’s confluent hypergeometric function \(W_{\lambda,\mu}(x)\) \({}^{(1)}\). As is known, the Hermite and Laguerre polynomials, as well as the Bessel functions, are special cases of this function.

We shall seek the solution of equation (1) in the form:

\[ u=\psi(x)F[z(x)], \tag{2} \]

where \(\varphi\), \(F\), and \(z\) are arbitrary functions. Substituting (2) into (1), we obtain:

\[ \left\{F''+\frac{1}{z}F'+F\left(1-\frac{s^2}{z^2}\right)\right\} +F'\left\{\frac{2\psi'}{\psi z'}+\frac{z''}{z'^2}-\frac{1}{z}\right\} + \]

\[ +F\left\{\frac{\psi''}{\psi z'^2}+\frac{f}{z'^2} -\left(1-\frac{s^2}{z^2}\right)\right\}=0, \tag{3} \]

where \( '\) denotes differentiation with respect to the argument, and the parameter \(s\) will be determined later. It should be noted that in (3) we have proceeded along the path of constructing Bessel’s equation for the function \(F(z)\). Equating to zero each of the expressions in braces, we obtain:

\[ u=\left(\frac{z}{z'}\right)^{1/2} \left\{AZ_s^{(1)}(z)+BZ_s^{(2)}(z)\right\}. \tag{4} \]

Here \(Z_s^{(1)}\) and \(Z_s^{(2)}\) are two linearly independent solutions of Bessel’s equation, \(A\) and \(B\) are constants, and \(z\) is determined by the equation

\[ z'^2(1+\varepsilon)=f^*, \tag{5} \]

where

\[ \varepsilon= \frac{z'''}{2z'^3} -\frac{3z''^2}{4z'^4} +\frac{1}{z^2}\left(\frac{1}{4}-s^2\right) -\frac{f-f^*}{z'^2}. \tag{6} \]

The parameter \(s\) and the arbitrary function \(f^*\) are chosen in such a way that over the entire interval of variation of \(x\) the quantity \(\varepsilon\) remains much less than unity. It can be shown that when \(f=f^*\) the asymptotic formulas are expressed through Bessel functions of order \(s=\pm \frac{1}{m+2}\), where \(m\) is the multiplicity of the zero of the equation \(f(x_0)=0\). Near the root \(x=x_0(1+\xi)\), where \(\xi\to0\),

the quantity

\[ \varepsilon \sim \frac{1}{\xi^{m+2}}\left[\frac{1}{m+2}-s^2\right]^{(2-4)}, \]

whence we find the condition under which \(\varepsilon\) remains much smaller than unity.

In particular, if \(x_0\) is a simple root of the equation \(f(x_0)=0\), the asymptotic solutions are expressed in terms of Bessel functions of order \(\pm 1/3\). Another choice of \(f^*\) ensures the condition \(\varepsilon \ll 1\) for a corresponding choice of the parameter \(s\), which makes it possible to obtain asymptotic formulas of Hilb type \((^5,^6)\). Neglecting \(\varepsilon\) in equation (5), we obtain for \(z\):

\[ z=\int (f^*)^{1/2}\,dx . \tag{7} \]

As is known, Whittaker’s equation has the form:

\[ \frac{d^2 W}{dx^2}+\left\{-\frac14+\frac{\lambda}{x}+\frac{1/4-\mu^2}{x^2}\right\}W=0. \tag{8} \]

Our task is to obtain asymptotic solutions of equation (8) for large \(\lambda\) and fixed \(\mu\), valid uniformly for all values of \(x\) from \(0\) to \(\infty\). Choosing \(f^*=-1/4+1/4\), it is necessary to consider three ranges of variation of \(x\):

\[ \begin{aligned} \text{I.}\quad &0\leq x\leq x^*<4\lambda, & z_1&=\int_0^x\left(\frac{\lambda}{x}-\frac14\right)^{1/2}dx, & s&=2\mu;\\ \text{II.}\quad &x^*\leq x\leq 4\lambda, & z_2&=\int_x^{4\lambda}\left(\frac{\lambda}{x}-\frac14\right)^{1/2}dx, & s&=1/3;\\ \text{III.}\quad &4\lambda\leq x<\infty, & z_3&=\int_{4\lambda}^x\left(\frac14-\frac{\lambda}{x}\right)^{1/2}dx, & s&=1/3. \end{aligned} \]

Here \(x^*\) is some interior point of the interval \(0,4\lambda\). According to (4), the asymptotic solutions for each of the three ranges take the form:

\[ W_{\lambda,\mu}(x)=\left(\frac{z_1}{z_1'}\right)^{1/2} \{A_1J_{2\mu}(z_1)+B_1N_{2\mu}(z_1)\}, \qquad 0\leq x\leq x^*<4\lambda, \tag{9} \]

\[ W_{\lambda,\mu}(x)=\left(\frac{z_2}{-z_2'}\right)^{1/2} \{A_2J_{1/3}(z_2)+B_2J_{-1/3}(z_2)\}, \qquad x^*\leq x\leq 4\lambda, \tag{10} \]

\[ W_{\lambda,\mu}(x)=\left(\frac{z_3}{z_3'}\right)^{1/2} \{A_3I_{1/3}(z_3)+B_3K_{1/3}(z_3)\}, \qquad 4\lambda\leq x<\infty. \tag{11} \]

The constants \(A_3\) and \(B_3\) are easily determined by using the behavior of \(W_{\lambda,\mu}(x)\), \(I_{1/3}(z_3)\), and \(K_{1/3}(z_3)\) at infinity, which gives:

\[ A_3=0,\qquad B_3=\frac{1}{\sqrt{\pi}}e^{-\lambda+\lambda\ln\lambda}. \tag{12} \]

Comparing (10) and (11) as \(x\to 4\lambda\), we obtain:

\[ A_2=B_2=\sqrt{\frac{\pi}{3}}\,e^{-\lambda+\lambda\ln\lambda}. \tag{13} \]

To determine the coefficients \(A_1\) and \(B_1\), we require equality of the asymptotic solutions (9) and (10) and their derivatives at some interior point of the interval \(0,4\lambda\), whence, taking into account that \(z_1+z_2=\pi\lambda\), we have

\[ A_1=\sqrt{\pi}\,e^{-\lambda+\lambda\ln\lambda}\sin(\lambda-\mu)\pi;\qquad B_1=-\sqrt{\pi}\,e^{-\lambda+\lambda\ln\lambda}\cos(\lambda-\mu)\pi. \tag{14} \]

Finally, the required asymptotic formulas take the form:

\[ W_{\lambda,\mu}(x)=\sqrt{\pi}\,e^{-\lambda+\lambda\ln\lambda} \left(\frac{z_1}{z_1'}\right)^{1/2} \{\sin(\lambda-\mu)\pi J_{2\mu}(z_1)-\cos(\lambda-\mu)\pi N_{2\mu}(z_1)\}, \tag{15} \]

\[ z_1=2\lambda\arcsin\left(\frac{x}{4\lambda}\right)^{1/2} +2\lambda\left(\frac{x}{4\lambda}\right)^{1/2} \left(1-\frac{x}{4\lambda}\right)^{1/2}, \qquad 0\leq x\leq x^*<4\lambda; \]

\[ W_{\lambda,\mu}(x)=\sqrt{\frac{\pi}{3}}\,e^{-\lambda+\lambda\ln\lambda} \left(\frac{z_2}{-z'_2}\right)^{1/2} \{J_{1/3}(z_2)+J_{-1/3}(z_2)\}, \]

\[ z_2=2\lambda\arccos\left(\frac{x}{4\lambda}\right)^{1/2} -2\lambda\left(\frac{x}{4\lambda}\right)^{1/2} \left(1-\frac{x}{4\lambda}\right)^{1/2},\qquad x^*\leq x\leq 4\lambda; \tag{16} \]

\[ W_{\lambda,\mu}(x)=\frac{1}{\sqrt{\pi}}\,e^{-\lambda+\lambda\ln\lambda} \left(\frac{z_3}{z'_3}\right)^{1/2}K_{1/3}(z_3), \]

\[ z_3=2\lambda\left(\frac{x}{4\lambda}\right)^{1/2} \left(\frac{x}{4\lambda}-1\right)^{1/2} -2\lambda\ln\left\{ \left(\frac{x}{4\lambda}\right)^{1/2} -\left(\frac{x}{4\lambda}-1\right)^{1/2} \right\},\qquad 4\lambda\leq x<\infty . \tag{17} \]

The arguments in formulas (15)—(17) can be simplified by introducing a new variable \(x=4\lambda\sin^2\frac{\varphi}{2}\) in (15), \(x=4\lambda\cos^2\frac{\varphi}{2}\) in (16), and \(x=4\lambda\operatorname{ch}^2\frac{\Phi}{2}\) in (17). Then (15)—(17) pass respectively into (18)—(20):

\[ W_{\lambda,\mu}\left(4\lambda\sin^2\frac{\varphi}{2}\right) =\sqrt{2\pi\lambda}\,e^{-\lambda+\lambda\ln\lambda} \left(\frac{\varphi+\sin\varphi}{\operatorname{ctg}(\varphi/2)}\right)^{1/2}\times \]

\[ \times\{\sin(\lambda-\mu)\pi J_{2\mu}(\lambda\varphi+\lambda\sin\varphi) -\cos(\lambda-\mu)\pi N_{2\mu}(\lambda\varphi+\lambda\sin\varphi)\}, \tag{18} \]

\[ 0\leq\varphi\leq\varphi^*<\pi; \]

\[ W_{\lambda,\mu}\left(4\lambda\cos^2\frac{\varphi}{2}\right) =\sqrt{\frac{2\pi\lambda}{3}}\,e^{-\lambda+\lambda\ln\lambda} \left(\frac{\varphi-\sin\varphi}{\operatorname{tg}(\varphi/2)}\right)^{1/2}\times \]

\[ \times\{J_{1/3}(\lambda\varphi-\lambda\sin\varphi) +J_{-1/3}(\lambda\varphi-\lambda\sin\varphi)\},\qquad 0\leq\varphi\leq\varphi^*<\pi; \tag{19} \]

\[ W_{\lambda,\mu}\left(4\lambda\operatorname{ch}^2\frac{\Phi}{2}\right) =\sqrt{\frac{2\lambda}{\pi}}\,e^{-\lambda+\lambda\ln\lambda} \left(\frac{\operatorname{sh}\Phi-\Phi}{\operatorname{th}(\Phi/2)}\right)^{1/2} K_{1/3}(\lambda\operatorname{sh}\Phi-\lambda\Phi), \tag{20} \]

\[ 0\leq\Phi<\infty . \]

If we assume that \(4\lambda\gg x^*\), (15) is simplified and takes the form

\[ W_{\lambda,\mu}(x)=\sqrt{2\pi x}\,e^{-\lambda+\lambda\ln\lambda} \{\sin(\lambda-\mu)\pi J_{2\mu}(2\sqrt{\lambda x})- \]

\[ -\cos(\lambda-\mu)\pi N_{2\mu}(2\sqrt{\lambda x})\}. \tag{21} \]

This is an asymptotic formula of Hilb type.

From formulas (15), as well as (18) and (21), it is seen that Whittaker’s equation will have a solution bounded at zero if the coefficient of \(N_{2\mu}(z)\) vanishes, i.e. \(\lambda=l+\mu+\frac12\), where \(l+1\) is a natural number. This makes it possible to determine exactly the eigenvalues of the energy operator of the hydrogen atom \(E_n=-\frac{mZ^2e^4}{2\hbar^2n^2}\), \(n=1,2,\ldots\) (see (22)) and of the harmonic oscillator \(E_n=(n+\frac12)\hbar\omega\), \(n=0,1,2,\ldots\) (see (26)).

Taking into account the relation of Laguerre polynomials with the Whittaker function

\[ L_l^\alpha(x)=(-1)^l x^{-\frac{\alpha+1}{2}}e^{x/2} W_{l+\frac{\alpha+1}{2},\,\frac{\alpha}{2}}(x), \tag{22} \]

we obtain the following formulas, valid for large \(l\):

\[ L_l^\alpha\left(4\lambda\sin^2\frac{\varphi}{2}\right) =\sqrt{\pi}\, \frac{\exp[-\lambda\cos\varphi+(\lambda-\alpha/2)\ln\lambda]} {(2\sin(\varphi/2))^\alpha}\times \]

\[ \times(\varphi\csc\varphi+1)^{1/2}J_\alpha(\lambda\varphi+\lambda\sin\varphi), \qquad 0\leq\varphi\leq\varphi^*<\pi; \tag{23} \]

\[ L_l^\alpha\left(4\lambda\cos^2\frac{\varphi}{2}\right) =(-1)^l\sqrt{\frac{\pi}{3}}\, \frac{\exp[\lambda\cos\varphi+(\lambda-\alpha/2)\ln\lambda]} {(2\cos(\varphi/2))^\alpha}\times \]
\[ {}\times(\varphi\csc\varphi-1)^{1/2} \left\{J_{1/3}(\lambda\varphi-\lambda\sin\varphi) +J_{-1/3}(\lambda\varphi-\lambda\sin\varphi)\right\}, \quad 0\leqslant\varphi\leqslant\varphi^*<\pi; \tag{24} \]

\[ L_l^\alpha\left(4\lambda\operatorname{ch}^2\frac{\Phi}{2}\right) =\frac{(-1)^l}{\sqrt{\pi}}\, \frac{\exp[\lambda\operatorname{ch}\Phi+(\lambda-\alpha/2)\ln\lambda]} {(2\operatorname{ch}(\Phi/2))^\alpha}\times \]
\[ {}\times(1-\Phi\operatorname{csch}\Phi)^{1/2} K_{1/3}(\lambda\operatorname{sh}\Phi-\lambda\Phi), \quad 0\leqslant\Phi<\infty. \tag{25} \]

In the last three formulas \(\lambda=l+\dfrac{\alpha+1}{2}\). The case of large \(\alpha\) and fixed \(l\) was considered in \((^7)\). For \(\alpha=\pm 1/2\), the Laguerre polynomials are simply related to the Hermite polynomials

\[ \mathscr{H}_{2l}(x)=(-1)^l2^{2l}L_l^{-1/2}(x^2), \qquad \mathscr{H}_{2l+1}(x)=(-1)^l2^{2l+1}xL_l^{1/2}(x^2), \tag{26} \]

and from formulas (23)—(25) we obtain

\[ H_n\left(\sqrt{2n+1}\sin\frac{\varphi}{2}\right) = \frac{(2n+1)^{n/2}}{(2\cos(\varphi/2))^{1/2}} \exp\left[-\frac{2n+1}{4}\cos\varphi\right]\times \]
\[ {}\times \cos\left\{\frac{2n+1}{4}(\varphi+\sin\varphi)-\frac{\pi n}{2}\right\}, \quad 0\leqslant\varphi\leqslant\varphi^*<\pi; \tag{27} \]

\[ H_n\left(\sqrt{2n+1}\cos\frac{\varphi}{2}\right) = \frac{1}{2}\sqrt{\frac{\pi}{3}}\,(2n+1)^{\frac{n+1}{2}} \exp\left[\frac{2n+1}{4}\cos\varphi\right] \left(\frac{\varphi-\sin\varphi}{\sin(\varphi/2)}\right)^{1/2}\times \]
\[ {}\times \left\{ J_{1/3}\left[\frac{2n+1}{4}(\varphi-\sin\varphi)\right] + J_{-1/3}\left[\frac{2n+1}{4}(\varphi-\sin\varphi)\right] \right\}, \quad 0\leqslant\varphi\leqslant\varphi^*<\pi; \tag{28} \]

\[ H_n\left(\sqrt{2n+1}\operatorname{ch}\frac{\Phi}{2}\right) = \frac{(2n+1)^{\frac{n+1}{2}}}{2\sqrt{\pi}} \exp\left[\frac{2n+1}{4}\cos\Phi\right] \left(\frac{\operatorname{sh}\Phi-\Phi}{\operatorname{sh}(\Phi/2)}\right)^{1/2}\times \]
\[ {}\times K_{1/3}\left[\frac{2n+1}{4}(\operatorname{sh}\Phi-\Phi)\right], \quad 0\leqslant\Phi<\infty. \tag{29} \]

In the case of the Laguerre polynomials, (21) passes into the well-known Hilb-type formula belonging to Szegő:

\[ L_l^\alpha(x)=\sqrt{2\pi}\exp\left[-\lambda+\lambda\ln\lambda+\frac{x}{2}\right]x^{-\alpha/2}J_\alpha(2\sqrt{\lambda x}), \tag{30} \]

and in the case of the Hermite polynomials one obtains Adamov’s formula

\[ H_n(x)=\frac{(2n+1)^{n/2}}{\sqrt{2}} \exp\left[-\frac{2n+1}{4}+\frac{x^2}{2}\right] \cos\left(\sqrt{2n+1}\,x-\frac{\pi n}{2}\right). \tag{31} \]

Moscow State University
named after M. V. Lomonosov

Received
23 V 1958

CITED LITERATURE

1. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 2, Moscow—Leningrad, 1934.
2. A. A. Sokolov, Vest. MGU, No. 4, 77 (1947).
3. D. Ivanenko, A. Sokolov, Classical Field Theory, Moscow—Leningrad, 1951.
4. R. E. Langer, Trans. Am. Math. Soc., 33, 23 (1931); 34, 447 (1932); R. E. Langer, Phys. Rev., 51, 669 (1937).
5. E. Hilb, Math. Zs., 5 (1919).
6. A. A. Sokolov, B. K. Kerimov, DAN, 108, No. 4, 611 (1956); A. Sokolov, B. Kerimov, Nuovo Cim., No. 5, 921 (1957); R. M. Muradyan, DAN, 115, No. 5, 887 (1957).
7. A. A. Sokolov, N. P. Klepikov, I. M. Ternov, ZhETF, 24, 249 (1953).