problems and historical background

$ (<*)»я+и(Р)т X^м yⁿ
2 —T Г “7 ’


(Y)m+« n\

he work of Humbert has been described reasonably fully by Appeli and Катрё de F^riet (1926, pp. 124-135), and the definitions and convergence conditions of all of these 20 confluent hypergeometric series in two variables are given also in Erddlyi et al. (1953, pp. 225-228). The definitions of Ф_ь Ф₂ and S₂, given in Erddlyi et al. [op. tit., p. 225, Equations (20), (21), and p. 226, Equation (26)] are in error; we first recall here the corrected definitions of Ф! and Ф₂:

H\y\<^w>
(
⁰ (Y)m+n nil n\

17) Ф₂[Р, P'; y; x, y] =
|x| < «>, |y| < oo,
so that
(18) Ф,[а, P; y; x, y] = lim F,[a, p, p'; y; x, уlP']
= lim Fi[a, p, 1/e; y; ey],
e-*0
and
(19) Ф₂[р, P'; _Y; x, y] = Urn Ffa p, p'_{; Y}; xhx, yla]
= jim Fjll/e, p, p'; y; ex, ey]

which follow, in view of 1.2 (26), from (2), (16) and (17).

F
(20)


Фз[Р; y; ■*» y] =


| (P)_W
т,л»о (y)_m+„ ml n\

or the sake of completeness, we also record here the definitions of similar confluent forms of the remaining Appel! series F_2t F₃ and F₄:
\x\ < ⁰⁰, |y| < °°;
*
Vi[a, p; Y. Y';jf» У]

(21)
(tt)m-*-n(P)m
m£- 0 (Y)m(Y')« "»! n!
M
(22)


^[a; у, y'; *, y] =


(23)


Ei[a, a', P; y; *, y]


~ (a)_m+„ X^я* у^л
2/ ’
"••ⁿ⁼⁰ (Y)m(Y')n w! n!
\x\ < |y| < oo;
= 2 —— ; ~⁵
"'•^л-° (Y)_m+«
И < i. Ы <

< i. M <
g
(24) S₂[a, p; y; •*, y]
(«UP)m У
^т-^л“° (у)л»+/. «!
W < !» Ы < ⁰⁰■
Notice that

25. 
    lim Ф₂[р, 1/e; y; at, ey] = Ф₃[р; у; at, y],
    

s—»0

25. 
    lim ЧМог, 1/e; y, y'; ex, y] = Ч^а; y, y'\*, y] and
    
26. 
    lim Si[a, 1/e, p; y; *, ey] = H₂[a, p; у; x, у].
    

E—»0
Катрё de Ftxiet’s'Series and Its Generalizations
Just as the Gaussian series ₂F_t was generalized to _pF_q by increasing the numbers of the numerator and denominator parameters, the four Appell series were unified and generalized by Катрё de Fdriet (1921) who defined a general hypergeometric series in two variables [see Appell and Катрё de Рёг1е1 (1926, p. 150, Equation (29))]. The notation introduced by Катрё de Feriet [loc. cit.] for his double hypergeometric series of
superior order was subsequently abbreviated by Burchnall and Chaundy (1941, p. 112). We recall here the definition of a more general double hypergeometric series (than the one defined by Катрё de Feriet) in a slightly modified notation! [see, for example, Srivastava and Panda (1976a, p. 423, Equation (26))]:
(
(28)
a_py.(b_q); Ы; ~| Ху у I
0/):(fU;(Y„); J
„ П («,),« П ( ft),Д (ft),
• П («,),„ П (ft), П (у,).^{r] j!}
/“• y-1 /-»
where, for convergence,
(0 P + 4 < l + m + 1, p + к < l + n + 1, |x| < «, \y\ < <*>,
or
(ii) p + q- l+ m+l, p + k = l + n + l, and
(29) { + \У\^1КР~° < 1.
[ max{|*|, |yj) < 1, if p < l.
Although the double hypergeometric series defined by (28) reduces to the Катрё de Рёг5е1 series in the special case:
q — к and m = n,
yet it is usually referred to in the literature as the Катрё de Рёг1е1 series.
A further generalization of the (so-called) Катрё de Feriet series (28) is due to Srivastava and Daoust (1969a) who indeed defined an extension of the _p4?_q series [cf. 1.2(38)] in two variables. More generally, in Section 1.4 we shall give the definition of an interesting multivariable extension of which is referred to in the literature as the generalized Lauricella series in several variables; it is due also to Srivastava and Daoust (1969b, p. 454).
We conclude this section by recording the following instances in which the Катрё de F6riet series (28) can be expressed in terms of generalized
t.Here, and elsewhere in this book, we find it convenient to abbreviate the array a, simply by (e,,), with similar interpretations for (b₉), el cetera.
[«

	Га„ ...	, a*:-;-;
(30)	F&	Xj
	Lp„ ...	> Pv*~
	Г-.«ь	... , (Xp-,yi, .
(31)
	L-:Pi,	..., P*;&i, ■

1. "] [Yi, - ,Vr; ]
pi P,; J ^rFs L -. J ’
Г
(32)
а„ ... _f oc_p:v;o; П Г а_ь ..., a_p> v + о; 1

„ П («,),« П ( ft),Д (ft), 32