UDC 517.516+517.944.1+517.947

MATHEMATICS

M. B. KAPILEVICH

ON DEGENERATE HYPERGEOMETRIC FUNCTIONS OF HUMBERT AND HORN

(Presented by Academician I. G. Petrovskii, 10 I 1966)

In the present work we consider the confluent hypergeometric series \(\Xi_2\) and \(H_3\) \((^1)\), which play an important role in the theory of the generalized wave equation \((^2,^3)\):

\[ Q[u]\equiv u_{\theta\theta}-u_{\sigma\sigma}-\frac{a}{\sigma}u_\sigma+c^2u=0. \tag{1} \]

For them the following results are established:

1. Let \(\beta>0,\ \lambda>0,\ \gamma=\delta-\beta+\mu-\lambda>0,\ c_0\Gamma(\beta)\Gamma(\lambda)\Gamma(\gamma)=\Gamma(\delta)\Gamma(\mu)\). Then

\[ H_3(\alpha,\beta,\delta;\,x,y)= \]

\[ =c_0\int_0^1 \xi^{\beta-1}(1-\xi)^{\gamma-1} F(\mu-\lambda,\delta-\lambda,\gamma;\,1-\xi) H_3(\alpha,\mu,\lambda;\,\xi x,y)\,d\xi . \tag{2} \]

In particular, for \(\lambda=\mu\),

\[ H_3(\alpha,\mu,\mu;\,x,y)=(1-x)^{-\alpha}\bar J_{-\alpha}[2\sqrt{y(1-x)}], \]

and therefore (2) gives

\[ H_3(\alpha,\beta,\delta;\,x,y) =c_1\int_0^1 \xi^{\beta-1}(1-\xi)^{\delta-\beta-1}(1-\xi x)^{-\alpha} \bar J_{-\alpha}[2\sqrt{y(1-\xi x)}]\,d\xi, \tag{3} \]

where \(\delta>\beta>0,\ c_1\Gamma(\beta)\Gamma(\delta-\beta)=\Gamma(\delta)\). The inverse of (3) is the equality

\[ (1-x)^{-\alpha}\bar J_{-\alpha}[2\sqrt{y(1-x)}] =c_2\int_0^1 \xi^{\lambda-1}(1-\xi)^{\mu-\lambda-1} H_3(\alpha,\mu,\lambda;\,\xi x,y)\,d\xi, \tag{4} \]

which arises from (2) when \(\delta=\beta,\ \mu>\lambda>0,\ c_2\Gamma(\lambda)\Gamma(\mu-\lambda)=\Gamma(\mu)\).

1. If \(a>0,\ b>0,\ \gamma=\alpha+\beta-a-b>0\), moreover \(\alpha\) and \(\beta\ne 0,-1,-2,\ldots\), then

\[ \Xi_2(a,b,c;\,x,y)= \]

\[ =\mu_1\int_0^1 \xi^{a-1}(1-\xi)^{\gamma-1} F(\alpha-b,\beta-b,\gamma;\,1-\xi)\Xi_2(\alpha,\beta,c;\,x\xi,y)\,d\xi, \tag{5} \]

where \(\mu_1\Gamma(a)\Gamma(b)\Gamma(\gamma)=\Gamma(\alpha)\Gamma(\beta)\). Conversely, putting \(\gamma_2>\gamma_1\ge 0\), we find

\[ \Xi_2(\alpha,\beta,\gamma_2;\,x,y_2) =\mu_2\int_0^1 \Xi_2(\alpha,\beta,\gamma_1;\,xt,y_1t)\, Q(y_1,y_2,t)\,dt, \tag{6} \]

\[ \mu_2\Gamma(\gamma_1)\Gamma(\gamma_2-\gamma_1)=\Gamma(\gamma_2), \]

\[ Q=t^{\gamma_1-1}(1-t)^{\gamma_2-\gamma_1-1} \bar J_{\gamma_2-\gamma_1-1}[2\sqrt{(y_1-y_2)(1-t)}]. \]

In the case \(y_1=0\), (6) gives an integral representation of the series \(\Xi_2\), and for \(y_2=0\) its inversion \({}_2F_1=U[\Xi_2]\) with respect to \(F(\alpha,\beta,\gamma_1;x)\). Composing the considered integral operators, we arrive at a series of combined relations. For example, if we insert (4) (with \(\alpha=\gamma_1-\gamma_2+1,\ x=t,\ y=y_1-y_2\)) into (6), or substitute \({}_2F_1=U[\Xi_2]\) into the known integral representations of the Appell functions \(F_k(x,y)\) \((k=1,2,3,4)\) (1), then as a result we arrive at relations connecting \(\Xi_2(x,y)\) with \(H_3(x,y)\) and \(F_k(x,y)\). The integrals (2)—(6) generate a number of infinite expansions for \(H_3\) and \(\Xi_2\). Thus, with the aid of (3) and Lommel’s multiplication theorem for \(\overline{J}_{-\alpha}[2\sqrt{y(1-x\xi)}]\), we find, in the interval \(|x|<1\),

\[ H_3(\alpha,\beta,\delta;x,y) = \sum_{n=0}^{\infty} \frac{(\beta)_n(xy)^n}{n!(1-\alpha)_n(\delta)_n} F(\alpha,\beta+n,\delta+n;x)\overline{J}_{\,n-\alpha}(2\sqrt{y}). \tag{7} \]

Starting from (2) and the multiplication theorem for \(H_3(\alpha,\mu,\lambda;\xi x,y)\), we arrive at the expansion of \(H_3(\alpha,\beta,\delta;x,y)\) in terms of the functions \(H_3(\alpha,\mu,\lambda-n;x,y)\), \((n=0,1,2,\ldots)\). Finally, replacing in (6) \(\overline{J}_{\nu}(2\sqrt{z})\) by the power series \({}_0F_1(\nu+1,-z)\), we obtain an addition theorem for \(\Xi_2\) in the argument \(y\). As a result of the limiting transition

\[ \lim_{\beta\to\infty}H_3(\alpha,\beta,\delta;x/\beta,y)=H_5(\alpha,\delta;x,y), \]

\[ \lim_{\alpha\to\infty}\Xi_2(\alpha,\beta,\gamma;x/\alpha,y)=\Phi_3(\beta,\gamma;x,y), \]

the preceding formulas pass into analogous equalities for \(H_5\) and \(\Phi_3\), among which we note only

\[ H_3(\alpha,\beta,\delta;x,y) = \frac{1}{\Gamma(\beta)} \int_{0}^{\infty} \xi^{\beta-1}e^{-\xi} H_5(\alpha,\delta;x\xi,y)\,d\xi \qquad (\beta>0). \tag{8} \]

Substituting (8), with \(\delta=\beta\), into the known integral representations of the functions \(\Phi_3(x,y)\), \(\Psi_2(x,y)\), and \(\Xi_2(x,y)\) (1), we connect these Humbert series with \(H_5(x,y)\).

Let us now consider several improper integrals with Bessel functions reducible to \(H_3\), \(H_5\), and \(\Xi_2\).

I. Denote by \(U_m(\beta,\mu,\nu)\) \((m=0,1,2,\ldots)\) the expression

\[ U_m=\int_{0}^{\infty} t^{\beta-2\nu-1} \overline{J}_{\mu}\!\left(a\sqrt{z^2+t^2}\right) J_{\nu+m}(bt)J_{\nu+m}(ct)\,dt \]

and put \(a^2>(b+c)^2,\ \nu>-1/2,\ -2m<\beta<\mu+2\nu+5/2,\ |\overline{\omega}|^2=b^2+c^2-2bc\cos\varphi\). Then we obtain

\[ U_m = D_m\frac{(bc)^\nu}{a^\beta} \int_{0}^{\pi} H_3\!\left( \frac{\beta}{2}-\mu,\frac{\beta}{2},\nu+1; \frac{\overline{\omega}^{\,2}}{a^2},\frac{a^2z^2}{4} \right) C_m^{\nu}(\cos\varphi)\sin^{2\nu}\varphi\,d\varphi, \tag{9} \]

if

\[ \pi\nu 2^{\,2-\beta}\Gamma(\mu+1-\beta/2)\Gamma(2\nu+m)D_m = m!\,\Gamma(\beta/2)\Gamma(\mu+1). \]

In particular, when \(\beta=2(\nu+1)\),

\[ U_0[2(\nu+1),\mu,\nu] = \frac{A_1(bc)^\nu}{a^{2\mu}R^{2(\nu-\mu+1)}} H_3\!\left( \nu-\mu+1,\nu+\frac12,2\nu+1; \frac{4bc}{R^2},\frac{z^2R^2}{4} \right), \tag{10a} \]

where \(R=\sqrt{a^2-(b-c)^2}\), \(A_1\Gamma(\nu+1)\Gamma(\mu-\nu)=2\Gamma(\mu+1)\). Conversely, assuming \((b-c)^2<a^2<(b+c)^2\), \(A_2\sqrt{\pi}\Gamma(\mu+1/2)=\Gamma(\mu+1)\), we arrive at the value

\[ U_0[2(\nu+1),\mu,\nu] = \frac{A_2R^{2\mu-1}}{a^{2\mu}\sqrt{bc}} \Xi_2\!\left( \frac12+\nu,\frac12-\nu,\mu+\frac12; \frac{R^2}{4bc},-\frac{z^2R^2}{4} \right). \tag{10b} \]

II. The function \(\Xi_2\) also arises in the study of the integral \(V_m(\beta,\mu,\nu)\) of the form

\[ V_m=\int_0^\infty t^{\beta-1}(z^2+t^2)^m \overline{J}_{\mu+m}\!\left(a\sqrt{z^2+t^2}\right) \overline{J}_{\mu+m}\!\left(b\sqrt{z^2+t^2}\right) \overline{J}_{\nu}(ct)\,dt. \]

For it, putting \(\omega=\sqrt{a^2+b^2-2ab\cos\varphi}\), \(c^2>(a+b)^2\), \(\mu>-1/2\), \(0<\beta<2\mu+\nu+5/2\), \(m=0,1,2,\ldots\), we obtain

\[ V_m=\frac{\delta_m}{(ab)^m c^\beta} \int_0^\pi \Xi_2\!\left(\frac{\beta}{2}-\nu,\frac{\beta}{2},\mu+1; \frac{\omega^2}{c^2},-\frac{z^2\omega^2}{4}\right) C_m^\mu(\cos\varphi)\sin^{2\mu}\varphi\,d\varphi, \tag{11} \]

\[ \pi\mu\Gamma(2\mu+m)\Gamma(\nu+1-\beta/2)\delta_m = 4^{\mu+m+\beta/2-1}m! \,\Gamma(\beta/2)\Gamma(\nu+1)\Gamma^2(\mu+m+1). \]

In particular, if \(m=0\), \(b=0\), \(\overline V_0=V_0|_{b=0}\), \(0<c<a\), \(0<\beta<\mu+\nu+2\), (11) gives

\[ \overline V_0=\varphi(\beta,\nu)c^{-\beta} \Xi_2(\beta/2-\nu,\beta/2,\mu+1;a^2/c^2,-{}^1\!/_{4}a^2z^2), \tag{12a} \]

where \(\varphi(\beta,\nu)=2^{\beta-1}\Gamma(\beta/2)\Gamma(\nu+1)/\Gamma(\nu+1-\beta/2)\). Conversely, from (9), for \(0<c<a\) it follows that

\[ \overline V_0=\varphi(\beta,\mu)a^{-\beta} H_3(\beta/2-\mu,\beta/2,\nu+1;c^2/a^2,{}^1\!/_{4}a^2z^2). \tag{12b} \]

III. With the series \(H_5\) there is associated the integral \(W_m(\mu,\nu)\) \((m=0,1,2,\ldots)\) of the form

\[ W_m=\int_0^\infty t^{\mu-1}e^{-t^2} J_{\nu+m}(at)J_{\nu+m}(bt)\, {}_0F_2(\mu/2,\nu+\mu/2;-yt^2)\,dt, \tag{13} \]

namely, under the conditions \(\nu>-1/2\), \(2(\nu+m)+\mu>0\), (13) reduces to

\[ W_m=\gamma_m(ab)^\nu \int_0^\pi e^{-\omega^2/4} H_5\!\left(1-\frac{\mu}{2},\nu+1;\frac{\omega^2}{4},y\right) C_m^\nu(\cos\varphi)\sin^{2\nu}\varphi\,d\varphi. \]

Here

\[ 4\pi\nu\Gamma(2\nu+m)\gamma_m=m!\,\Gamma(\nu+\mu/2), \]

and, when \(\mu=2\),

\[ W_m(2,\nu)=\frac12\exp\!\left(-\frac{a^2+b^2}{4}\right) J_0(2\sqrt{y})J_{\nu+m}({}^1\!/_{2}ab). \]

In particular, \(\lim_{b\to0}[b^{-\nu}W_0]\) gives the integral representation

\[ H_5= \frac{e^x}{\Gamma(\delta-\alpha)} \int_0^\infty t^{\delta-\alpha-1}e^{-t}\, {}_0F_2(1-\alpha,\delta-\alpha;-yt)\, \overline{J}_{\delta-1}(2\sqrt{xt})\,dt, \]

from which, for \(\delta>\alpha>0\), \(\alpha\ne1,2,\ldots\), one obtains asymptotic estimates characterizing the behavior of the function \(H_5\) as \(x\to\infty\) and \(y\to\infty\). With the aid of (10) and (12) one can construct a number of important solutions of equation (1). For example, denote by \(H(\theta,\sigma,\theta_0,\sigma_0)\) and \(\overline H(\theta,\sigma,\theta_0,\sigma_0)\) the Hadamard functions of two singular Tricomi problems considered in \({}^{(2)}\), and let \(V(\beta)=(2\sigma_0)^{-\alpha}H\), \(\overline V(\beta)=(2\sigma_0)^{-\alpha}\overline H\) \((\alpha=2\beta)\). Then, relying on (10a), we obtain:

\[ V(\beta)=(4\sigma\sigma_0)^{1-\alpha}\overline V(1-\beta), \qquad \overline V(\beta)=\tilde{x}R^{-\alpha}H_3(\beta,\beta,2\beta;\omega,\rho), \tag{14} \]

where \(\omega=4\sigma\sigma_0/R^2\), \(\rho={}^1\!/_{4}c^2R^2\), \(R=\sqrt{(\theta-\theta_0)^2-(\sigma-\sigma_0)^2}\), \(\tilde{x}\Gamma(1-\beta)\Gamma(\alpha)=\Gamma(\beta)\). Since the functions (14) possess logarithmic singularities on the characteristics \(\theta\pm\sigma=\theta_0\pm\sigma_0\), they simultaneously give fundamental (elementary) solutions of equation (2). From (10a) there also arise integrals of equation (1)

\[ V_1(\beta)=(\sigma\sigma_0)^{1-\alpha}\overline V_1(1-\beta), \qquad \overline V_1(\beta)=(\theta-\theta_0)R^{-\alpha-2}H_3(\beta+1,\beta,2\beta;\omega,\rho), \tag{15} \]

which have singularities of a different character on the lines \(R=0\). Finally, (10b) generates the functions \(\left(R_1=\sqrt{(\theta-\theta_0)^2-(\sigma+\sigma_0)^2}\right)\)

\[ U=(\sigma\sigma_0)^{-\beta}\Xi_2(\beta,1-\beta;1;1/\omega,-\rho), \]

\[ U_1=(\sigma\sigma_0)^{-\beta}\Xi_2(\beta,1-\beta,1-R_1^2/4\sigma\sigma_0,-{}^1\!/\!_4c^2R_1^2), \]

considered earlier in \((^2,^3)\) (\(U_1\) was denoted in \((^2)\) by \(u_5\)). Another important class of particular solutions of equation (1) arises from (12) \(\left(t=\theta^2/\sigma^2,\ \xi={}^1\!/\!_4c^2\sigma^2\right)\):

\[ \begin{aligned} u_1^{(\nu)}&=\theta^\nu\Xi_2(-\nu/2,(1-\nu)/2,\beta+{}^1\!/\!_2;1/t,\xi),\\ u_2^{(\nu)}&=\theta^{-\nu}r^{2\nu-a}\Xi_2(\nu/2,(1+\nu)/2,\nu-\beta+1,r^2/\theta^2,-c^2r^2/4),\\ u_3^{(\nu)}&=\sigma^\nu H_3((1-\nu-a)/2,-\nu/2,{}^1\!/\!_2;t,-\xi),\\ u_4^{(\nu)}&=\sigma^\nu\sqrt{t}\,H_3((2-a-\nu)/2,(1-\nu)/2,{}^3\!/\!_2;t,-\xi),\quad r=\sqrt{\theta^2-\sigma^2}. \end{aligned} \tag{16a,b} \]

Since, together with \(u(\theta,\sigma,a)\), equation (1) is also satisfied by \(\bar u=\sigma^{1-a}u(\theta,\sigma,2-a)\), in turn (16) generate four other integrals \(\bar u_k^{(\nu)}\) \((k=1,2,3,4)\). The functions \(u_k^{(\nu)}, \bar u_k^{(\nu)}\), \((k=3,4)\), as well as (10a), (12b), (14), and (15), are transformed by the formula
\[ H_3=(1-x)^{-\alpha}H_3[\alpha,\delta-\beta,\delta;x/(x-1),y(1-x)], \]
while (10a), (14), and (15) are brought to another form by means of the quadratic transformation

\[ H_3(\alpha,\beta,2\beta;x,y)=\left(\frac{2}{2-x}\right)^\alpha H_{10}\left[\alpha,\beta+{}^1\!/\!_2;{}^1\!/\!_4\left(\frac{x}{2-x}\right)^2,{}^1\!/\!_2y(2-x)\right]. \]

Let us note that the series
\[ u=\sum_{n=0}^{\infty}[a_nu_1^{(n)}+\bar a_n\bar u_1^{(n)}], \]
composed of the functions (16a), solves the singular Cauchy problem
\[ u(\theta,0)=\sum_{n=0}^{\infty}a_n\theta^n,\qquad u_\eta(\theta,0)=\sum_{n=0}^{\infty}\bar a_n\theta^n, \]
and \(u_4^{(0)}\) and \(\bar u_4^{(0)}\) give the Riemann functions of this problem \((^2)\).

For \(c=0\), \(V(\beta)\), \(\bar V(\beta)\), \(u_k^\nu\), and \(\bar u_k^\nu\) \((k=1,2,3,4)\) turn into Hadamard functions and the well-known self-similar (homogeneous) solutions of the Euler—Poisson equation \((^4,^5)\).

In the iteration method indicated in \((^2)\), \(u_k^{(\nu)}\) and \(\bar u_k^{(\nu)}\) play the role of the first approximation to analogous solutions of the Chaplygin equation \(T[u]=0\); moreover, as shown in \((^2)\), the subsequent approximations also belong to the class considered here of degenerate hypergeometric functions of Horn and Humbert. Therefore the relations found for \(\Xi_2\) and \(H_3\) also give the corresponding properties of the principal integrals of the equations \(Q[u]=0,\ T[u]=0\) listed above.

Moscow Evening
Metallurgical Institute

Received
6 I 1966

CITED LITERATURE

\(^1\) A. Erdélyi, Higher Transcendental Functions, 1, 1953.
\(^2\) M. B. Kapilevich, DAN, 81, No. 1 (1951); 91, No. 4 (1953); 154, No. 2 (1964).
\(^3\) P. Henrici, Zs. ang. Math. u. Phys., 8, No. 3, 169 (1957).
\(^4\) S. Gellerstedt, Ark. mat., astr. och fys., 25A, No. 29 (1937).
\(^5\) F. I. Frankl, DAN, 56, No. 7, 683 (1947).