$=B(\alpha, \beta)\int^{\infty}\phi(z)z_{p+*+k}^{a+\beta-1}F_{q+s+k}[b_{1}^{1},\cdot b^{t},\Delta(s+k,\alpha+\beta);a,\cdot.\cdot.\cdot,' a\ovalbox{\tt\small REJECT} ...
CERTAIN DOUBLE WHITTAKER TRANSFORMS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS By
H. M. SRIVASTAVA and C. M. JOSHI (Received September 30, 1967)
1. INTRODUCTION. Rainville [16, p. 104], and Abdul-Halim and Al-Salam [1] have shown that the single and double Euler transformations of the hypergeometric function $pFq$ are effective tools for augmenting its parameters. The double Laplace transforms of the $pFq$ -function, given by Singh [17] and Jain [14], have similar interesting properties. Recently, Srivastava and Singhal [20] have discussed a double Meijer transform of the generalized function in the form $hyperg\ovalbox{\tt\small REJECT} metric$
(1.1}
$\int_{0}^{\infty}\int_{0}^{\infty}\cdot v\lambda(x+y)$
.
$pFq[_{b_{I},b_{q};}^{a_{1},\cdot.\cdot.\cdot.a_{p};}tx^{s}y^{k}(x+y)^{r}]dxdy$
.
$=\frac{\sqrt{}\overline{\pi}\Gamma(\alpha+\beta+\mu\pm\nu+1z)}{\lambda^{\alpha*\beta+\mu}+\doteqdot\Gamma(\alpha+\beta+\mu+1)}B(\alpha, \beta)_{p+\S s+3\backslash +2r}\prime a_{1},\cdot.\cdot.\cdot$ $’ a_{b_{Q}^{p}’}b_{1},,$
,
$\Delta(S+k,\alpha+\beta),\Delta(S+k,\alpha+\beta\Delta(s_{\ddagger_{\mu}^{\beta}\ddagger^{\mu\pm\nu+z});}\alpha),\Delta(k,\beta^{1},\Delta(S+k+r, \alpha 1J;1t\delta(\frac{s+k+r}{\lambda})^{s+k+r}]$
where
$r,$ $s,$
$k$
are non-negative integers; $R_{e}(a)>0,$ $R_{e}(\beta)>0,$ stands for the set of parameters
$R_{e}(\lambda)>0$
$R_{e}(\alpha+\beta+\mu\pm\nu+\doteqdot)>0;\Delta(s, \alpha)$
$\div$
and
$s$
$\frac{\alpha+1}{s}$
$\frac{\alpha+s-1}{s}$
$\frac{\alpha+2}{s}$
and for the sake of brevity, the gamma-product $\Gamma(\alpha+\beta)\Gamma(\alpha-\beta)$
is denoted by $\Gamma(\alpha\pm\beta)$
,
and $\delta=\frac{s^{:}k^{k}}{(s+k)^{l+\mathcal{L}}}$
.
In this paper we present a double Whittaker transform of the generalized hypergeometric functions which leads to yet another interesting process of augmenting
H. M. SRIVASTAVA and C. M. JOSHI
18
the parameters in the $pFq$ -function. As a first instance of the usefulness of the generalized operator which we introduce in the next section, we give an alternative derivation of Kummer’s theorem and Weisner’s bilateral generating function, and discuss its numerous other applications to certain classical polynomials and Appell’s functions. In the various formulae the parameters, occurring in the operators employed, are restricted for reasons of $\omega nvergence$ and $term- by\cdot term$ integration.
2. THE GENERAL FORMULAE. We start with the following known result [10, p. 177]
$\int^{\infty}\int^{\infty}\phi(x+y)x^{\alpha-1}y^{\beta-1}dxdy=B(\alpha, \beta)\int^{\infty}\phi(z)z^{\alpha+\beta-1}dz$
which holds when
and
$R_{e}(\alpha)>0$
$ R_{e}t\beta$
) $>0$
.
Following the technique of term-by-term integration it is easy to prove that, if are non-negative integers, then inside the region of $R_{e}(a)>0,$ $R_{e}(\beta)>0$ and if convergence of the resulting series $r,$ $s,$
(2. 1)
$k$
$\int^{\infty}\int^{\infty}\phi(x+y)x^{\alpha-1}y^{\beta-1_{p}}F_{q}[b_{1},,b_{q}^{p};tx^{s}y^{k}(x+y)^{r}]dxdy$
$=B(\alpha, \beta)\int^{\infty}\phi(z)z_{p+*+k}^{a+\beta-1}F_{q+s+k}[b_{1}^{1},\cdot b^{t},\Delta(s+k,\alpha+\beta);a,\cdot.\cdot.\cdot,’ a\ovalbox{\tt\small REJECT},\Delta(s,\alpha),\Delta(k,\beta);t\delta z^{\iota+k+r}]dz$
,
where, as before, $\delta=\frac{s^{\prime}k^{k}}{(s+k)^{+k}}$
.
In particular, if we let $\phi(z)=e^{-\lambda z}z^{\sigma}E[c, d:
:
\lambda z]$
in terms of $MacRobert’ sE$-function, and invoke the formula [15, p. 396] $\int^{\infty}e^{-at}t^{b-1}E$
where (2. 2)
$R_{e}(a)>0,$
[
$c,$
$d:$
: at]
$dt=\frac{\Gamma(c)\Gamma(d)\Gamma(b+c)\Gamma(b+d)}{a^{b}11(b+c+d)}$
$R_{e}(b+c>0, R_{e}(b+d)>0$ ,
we obtain
$\int^{\infty}\int^{\infty}x^{a-1}y^{\beta-1}e^{-\lambda(x+y)}(x+y)^{\sigma}E[c, d:
; \lambda(x+y)]pF_{(J}[a_{1}b_{1}’\cdot.\cdot.\cdot.’
$=\underline{\Gamma(c)f’}\frac{(d|\Gamma(\alpha+\beta+}{\lambda^{a+\beta+\sigma}\Gamma(\alpha+}\frac{\sigma+c)\Gamma(\alpha+\beta+\sigma+d)}{\beta+\sigma+c+d)}B(\alpha, \beta)$
a_{p}b_{q};;tx^{s}y^{k}(x+y)^{r}]dxdy$
.
$p+3s+3k+\cdot rFq+\circ s+2k+r[^{a_{b_{1}^{1}},\cdot.\cdot.\cdot.’ a_{p},\Delta(s,\alpha),\Delta(k,\beta),\Delta(s+k+r,\alpha+\beta+\sigma+d);}$ $b_{q},$ $\Delta(s+k, \alpha+\beta),$ $\Delta(s+k+r, \alpha+\beta+\sigma+c+d)$
;
$t\delta\delta^{\prime}]$
,
CERTAIN $DOoe3LE$ WHITTAKER TRANSFORMS
provided the double integral $R_{e}(\alpha+\beta+\sigma+d)>0$
,
$\ovalbox{\tt\small REJECT} nverges,$
19
$R_{e}(a+\beta+\sigma+c1>0$ ,
$R_{e}(\alpha)>0,$ $R_{e}(\beta)>0,$ $R_{e}(\lambda)>0,$
and $\delta^{\prime}=(\frac{s+k+r}{\lambda})^{\iota+k+r}$
Since [11, p. 205] $E[z^{1}-\mu-\nu, \doteqdot-\mu+\nu; :
z]=l^{\urcorner}(\#-\mu\pm\nu)z^{-\mu}e^{t}z^{z}W_{\mu}$
,
.
$(z)$
,
in terms of Whittaker function, he formula (2. 2) shows that inside the region of convergence of the double integral, (2. 3)
$\int^{\infty}\int^{\infty}x^{x-1}y^{\beta-1}(x+y)^{\sigma}e^{-J}z^{\gamma(xty)}W_{\mu’\nu}[\lambda(x+y)]\cdot F[a_{1}b_{1}’\cdot.\cdot.\cdot.’
a_{p}b_{q};;tx^{s}y^{k}(x+y)^{r}]dxdy$
$=_{\frac{\Gamma(\alpha+\beta+\sigma\pm\nu+z^{1})}{\lambda^{a+\beta+\sigma}\Gamma(\alpha+\beta+\sigma-\mu+1)}B(\alpha,\beta)\cdot p+S\iota+31+21F_{q+2s+2k+r}[}a_{1},\cdot.\cdot.\cdot,$
$a_{p},\Delta(s,\alpha),\Delta(k, \beta)b_{1},\cdot,b_{q},\Delta(s+k,\alpha+\beta),$
,
$\Delta(s+k+r, \alpha+\beta+\sigma\pm\nu+\tau 1);t\delta\delta^{\prime}]$
$\Delta(s+k+r, a+\beta+\sigma-\mu+1)$
where $\mathfrak{T}1)>0$
are non-negative integers, $R_{e}(\alpha)>0,$ $R_{e}(\beta)>0,$ , and the resulting hypergeometric series converges. $r,$ $s,$
$k$
When
$\mu=0$ ,
our formula (2. 3) corresponds to $W_{0,\mu}(z)=[\frac{z}{\pi}]^{1}zK_{\mu}[\frac{1}{2}z]$
and if in addition, we set
(1. 1),
$R_{e}(\lambda)>0,$
;
$R_{e}(\alpha+\beta+\sigma\pm\nu+$
since
;
and make use of the relation
$\nu=\pm\doteqdot$
$K+z1(z)=[\frac{\pi}{2z}]^{2}e^{-z}1$
the special case
$\sigma=-*$
of (2. 3) will lead us to the earlier results of Jain [14] and Singh
[17].
In terms of the operator (2.4)
$\Omega_{(a,\beta.\sigma)}^{(\mu,\nu)}$
$\{$
$\}\equiv_{\Gamma(+++\overline{\doteqdot})}^{\Gamma(+++1)}--\frac{\alpha}{\alpha}\frac{\beta}{\beta}\frac{\sigma-\mu}{\sigma\pm\nu}--[B(\alpha, \beta)]^{-1}\cdot\int^{\infty}\int^{\infty}x^{a-1}y^{\beta-1}(x+y)^{\sigma}e^{-}z^{1_{(x+y)}}$
. . $W_{\mu}$
(2. 3) with (2. 5)
$\lambda=1$
and
$r=0$
$\}dxdy$ ,
$\{$
assumes the form
$\Omega_{(\alpha,\beta.\sigma)}^{(\mu,\nu)}\{F[_{b_{1},,b_{q}^{p};}^{a_{1},\cdot.\cdot.\cdot.’
.
$’ a_{l/},$
a;}tx^{l}y^{k}]\}$
$\Delta(s, \alpha),$
$\Delta(k, \beta),$
$=_{p+3s+3k}F_{q+2\$+2k}[a_{1}b_{1}’\cdot.\cdot.\cdot$
and it follows that
, $[x+y]$
$\Delta(s+k, \alpha+\beta+\sigma\pm\nu+\not\in);ts^{s}k^{k}]$ $b_{1},,$
$\Delta(s+k, \alpha+\beta),$
$\Delta(s+k, \alpha+\beta+\sigma-\mu+1)$
;
,
H. M. SRIVASTAVA and C. ’M. JOSHI
20 (2.6)
$q_{\alpha,\beta,\sigma)\{}^{0,\nu)}$
where [20,
$\Omega_{(\alpha,\beta.\mu)}^{(\nu)}$
}
$\equiv R_{a.\beta\sigma)\{}^{v)}$
}
denotes the operator introduced earlier by Srivastava and Singhal
\S 1]. The following operational relations are immediate
(2. 7)
$\Omega_{(a,\beta,\sigma)}^{(\mu,\nu)}\{1\}=1$
(2. 8)
.
$\Omega_{(a.\beta,)}^{(\mu,\nu)}\{e^{xt}\}=s^{F_{2}}[_{\alpha+\beta,\alpha}^{\alpha,\alpha+\beta}\ddagger^{\sigma\pm\nu+\doteqdot;}\beta+\sigma-\mu+1;^{t}]$
(2. 9)
of 2. $5):-$ $($
$\ovalbox{\tt\small REJECT} nsequences$
.
$f2_{(\alpha}^{(\mu\nu)}:_{\beta,\sigma)}\{lF_{0}[a;-;xt]\}=s^{F_{2}}[_{\alpha+\beta,\alpha+\beta+\sigma-\mu+}^{a,\alpha,\alpha+\beta+\sigma\pm\nu+\doteqdot}i_{;}t]$
,
a being a non-positive integer. (2.
10)
(2. 11) $(2, 12)$
$\Omega_{(\alpha.\beta.\sigma)}^{(\mu,\nu)}\{F[a;b;xt]\}=4F_{3}[_{b,a+\beta,\alpha}^{a,\alpha,\alpha+\beta}\ddagger_{\beta+\sigma-\mu+1;^{t}}^{\sigma\pm\nu+\doteqdot;}]$
$\Omega_{(aB,\sigma)}^{(\mu,.\nu)}\{1F_{1}[a;b;t(x+y)]\}=3F_{2}[_{b,\alpha}^{a,\alpha}\ddagger^{\beta}\beta\ddagger_{\sigma-\mu}^{\sigma\pm\nu}\ddagger^{1}\tau 1;;t]$
$\Omega_{(a};_{\beta.\sigma)}1_{0}F_{1}[-;a;xt]$
}
$=\S F_{S}[a\alpha,$
.
.
$\alpha\alpha\ddagger^{\beta+\sigma\pm\nu}\beta,\alpha+\beta\ddagger_{\sigma-\mu+1;}^{\doteqdot;}t]$
.
(2. 13)
$\Omega_{(a}^{(\mu}:_{\beta.\sigma)}^{\nu)}\{oF_{1}[-;a;t(x+y)]\}=2F_{2}[_{a,\alpha+\beta+\sigma-\mu^{Z}+1;}^{\alpha+\beta+\sigma\pm\nu+;}1t]$
(2. 14)
$\Omega_{(\alpha,\beta.\sigma)}^{(\mu,\nu)}\{2F_{1}[a, b;c;xt]\}=\iota^{F_{8}}[_{c,\alpha+\beta,\alpha+\beta+\sigma-\mu+i;}^{a,b,\alpha,\alpha+\beta+\sigma\pm\nu+\not\in}t]$
where a or (2. 15) (2. 16)
$b$
.
,
is a non-positive integer.
$\alpha+\beta+\sigma\pm\nu+\xi;]$ $\alpha+\beta+\sigma-\mu+1;tb,\alpha,$ $\Omega_{(a,\beta,\sigma)}^{c\mu,\nu)}\{2F_{2}[a, b;c, d;xt]\}=\iota^{F}\iota[ca,d,\alpha+\beta,$
$\Omega_{(\alpha}^{(\mu\nu)};_{\beta.\sigma)}\{x^{m}y^{n}(x+y)^{p}\}=\frac{(\alpha)_{m}(\beta)_{n}(\alpha+\beta+\sigma\pm\nu+T1)_{m+n+n}}{(\alpha+\beta)_{n+n}(\alpha+\beta+\sigma-\mu+1)_{m+n+p}}$
.
.
In particular, we shall need the following results. $\not\in(\alpha^{\prime}+\mu+\nu+1);-4a]$
(2. 17)
$\Omega,z(a^{\prime},\beta^{\prime}-1)\{(ax)^{\mu+\nu-\lambda}J_{\lambda}(2ax)\}=\frac{a^{\mu+\nu}(\alpha^{\prime})_{\mu+\nu}}{1^{\urcorner}(\lambda+1)}2F1[\not\in_{\lambda+1;}(\alpha^{\prime}+\mu\dashv\nu),$
(2. 18)
$\Omega_{(a^{\prime},\beta^{\prime}.-1)}^{(1^{1})}z\{J_{\mu}(ux)J_{\nu}(vx)\}=\frac{\Gamma(\alpha^{\prime}+\mu+\nu)}{l’(,.\alpha^{\prime})1’(\mu+1)1’(\nu+1)}(\not\in u)^{\mu}(\neq v)^{\nu}$
.
$F_{4}[*(\alpha^{\prime}+\mu+\nu), \doteqdot(\alpha^{\prime}+\mu+\nu+1) ; \mu+1, \nu+1 ; -u\sim, -v^{2}]$
.
,
where $F$ is Appell’s double hypergeometric function of the fourth type defined by means of [18, p. 211] $F_{4}[a, b ; c, c^{\prime} ; x,y]=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\underline{(a})(c^{\frac{n+\eta(b)_{m+n}}{)_{m}(c^{\prime})_{n}}\frac{x^{v\iota}}{m!}\frac{v^{n}}{n!}}$
CERTAIN OOUBLE WHITTAKER TRANSFORMS
provided, for convergence,
$|x|\S+|y|^{\frac{1}{2}}
