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\beginreview
\author Andrews, George S.

\title On $q$-analogues of the Watson and Whipple summations.

\citation \sl SIAM J. Math.\ Anal. \bf 7 \rm (1976), no. 3, 332--336.

\reviewer W. A. Al-Salam (Edmonton, Alta.)

\text The author proves two summation theorems for $q$-series: If $b=q↑{-n}$, then
$$\twoline{↓4\phi↓3\left[{a,\atop }\,{b,\atop(abq)↑{1/2},}\,
{c↑{1/2},\atop-(abq)↑{1/2},}\,{-c↑{1/2};\atop c;}\,{q,q\atop}\right]}{
={a↑{n/2}(aq;q↑2)↓∞(bq,q↑2)↓∞(cq/a;q↑2)↓∞(cq/b;q↑2)↓∞\over
(q;q↑2)↓∞(abq;q↑2)↓∞(cq;q↑2)↓∞(cq/ab;q↑2)↓∞}}$$
and if $a=q↑{-n}$, then
$$\twoline{↓4\phi↓3\left[{a,\atop}\,{q/a,\atop -q}\,{c↑{1/2},\atop e,}
{-c↑{1/2};\atop cq/e;}
\,{q,q\atop}\right]}{={\dispstyle{
q↑{{1\over2}n(n+1)}(ea;q↑2)↓∞(eq/a;q↑2)↓∞(eq/a;q↑2)↓∞\qquad\atop\hfill
(caq/e;q↑2)↓∞(cq↑2/ae;q↑2)↓∞}\over(e,q)↓∞(cq/e;q)↓∞}.}$$
The first is a $q$-analog of Watson's theorem which sums the hypergeometric
series $↓3F↓2(a,b,c/2;{1\over2}(a+b+1),c;1)$ and the second is a $q$-analog of
Whipple's theorem which sums $↓3F↓2(a,b,{1\over2}c; d,e;1)$, where
$a+b=1$, $d+e=1+c$.

\endreview

\beginreview
\author Marino, Mario

\title Un risultato di regolarit\`a della soluzione del problema di Cauchy
per certe equazioni differenziali del secondo ordine in spazi di
Hilbert.\xskip(English summary)

\citation \sl Ricerche Mat.\ \bf 24 \rm (1975), no.\ 1, 152--171.

\reviewer Angelo Favini (Bologna)

\text If $K$ is a complex Hilbert space, $0<\theta<1$,\xskip$u(\cdot)$ a $K$-valued
function defined in $\mathopen]-∞,T\mathclose[$, let $[u]↓\theta↑2$ be the integral
over $\mathopen]-∞,T\mathclose[\times\mathopen]-∞,T\mathclose[$ of the function
$(t,\xi)\mapsto|t-\xi|↑{-1-2\theta}\|u(t)-u(\xi)\|↓K$; then the space $H↑\theta
(-∞,T;K)$ consists of all $u\in L↑2(-∞,T;K)$ with $[u]↓\theta<+∞$, endowed with
the norm $\|u\|↑2↓{H↑\theta}=\|u\|↓{L↑2}↑2+[u]↓\theta↑2$; if $1<\theta<2$,\xskip
$H↑\theta(-∞,T;K)=\leftset u\in H↑1(-∞,T;K)\mathrel: u↑\prime\in H↑{\theta-1}(-∞,
T;K)\rightset$, with norm $\|u\|↑2↓{H↑\theta}=\|u\|↑2↓{H↑1}+[u↑\prime]↑2↓{\theta
-1}$. Let $V$, $H$ be separable complex Hilbert spaces,\xskip$V$ densely imbedded
in $H$, with norms $\|\ \|$, $|\ |$, respectively; suppose that $a(t;u,v)$,\xskip
$0≤t≤T<+∞$, is a sesquilinear form on $V\times V$ satisfying the following
conditions:\xskip(1) $a(t;u,v)=q(u,v)+r(t;u,v)$ for $u,v\in V$;\xskip(2) $q(u,v)$
is a sesquilinear form continuous on $V\times V$,\xskip
$q(u,v)=\overline{q(v,u)}$,\xskip$q(v,v)≥C\|v\|↑2$,\xskip$C>0$;\xskip
(3) $|r(t;u,v)|≤C↓1|u|\cdot|v|$,\xskip$0≤t≤T$;\xskip(4) $|r({t↑\prime;u,v)-
r(t↑{\prime\prime};u,v)}|≤N|{t↑\prime-t↑{\prime\prime}}|↑α|u|\cdot|v|$ for $t↑\prime,
t↑{\prime\prime}\in[0,T]$. Under these conditions the author shows that if
$f(t)\in H↑\theta(-∞,T;H)$,\xskip$0<\theta<α$,\xskip$f(t)=0$ for $t<0$, then the
solution $u(\cdot)$ of the problem $a(t;u(t),v)+\mu(u↑\prime(t),v)+(d/dt)\biglp u↑
\prime(t),v\bigrp=\biglp f(t),v\bigrp$, for all $v\in V$,\xskip$\mu\in\Cbf$,\xskip
$u(t)\in L↑2(-∞,T;V)$, $u↑\prime(t)\in L↑2(-∞,T;H)$,\xskip$u(t)=0$ for $t<0$,
belongs to $H↑\theta(-∞,T;V)∩H↑{1+\theta}(-∞,T;H)$ and its norm is estimated
in terms of $f$. Corresponding results for first order equations were obtained
by {\nm S. Campanato} [see Boll.\ Un.\ Mat.\ Ital.\ (4) \bf 6 \rm(1972),
112--121; \MR{48\#656}; ibid. (4) {\bf 6} (1972), 131--133; \MR{48\#657}].

\endreview