= ZZ [(qjw(ft)u(/3/)l*"y'

Expansions Involving Hypegeometic Functions of Two Vaiables By Aun Vema 1. Intoduction. A systematic study of expansions of Appell 's hypegeometic functions of two vaiables has been made by Buchnall and haundy [3]-[5] in 1942. Recently, Ragab [11], obtained two expansions of Kampé de Féiet's double hypegeometic functions which besides incopoating some of the expansions of Buchnall and haundy [3], [4], gave some new expansions also. In 2 of this pape a systematic attempt has been made to extend the expansions concening Appell functions to the Kampé de Féiet's double hypegeometic functions by using the two symbolic opeatos of Buchnall and haundy. In 4 using the method of iteation of seies some of the expansions due to haundy [5], [6], Niblett [10], Wimp and Luke [8] have been extended to the Kampé de Féiet's function. The pape is concluded by showing how the induction by using the Laplace tansfom and its invese can be employed to extend these esults to G-functions of two vaiables defined ecently by Agawal [1]*. It may be pointed out that these expansions ae vey geneal in natue and incopoate a vey lage numbe of expansions fo the functions of two vaiables. 2. Kampé de Féietf [2] intoduced the double hypegeometic functions of highe ode (i.e. with moe paametes) in two vaiables, namely: (1) («J (ft); (/?/) («-); (O = ZZ [(qjw(ft)u(/3/)l*"y' [Ik [1]» [ W [(«.)]» [(«.')]. ' povided p + v < p + a + 1 o p + v = p + c + 1 and x j + \y \ < min (1, 2"-"+I). Hee (aa) means A paametes of the type ax, a2,, aa and [a]n = a[a + l][a + 2] [a + n - 1]; [o]0 = 1. Fo special values of p, v, p, a the function (1) educes to the Appell's fou double hypegeometic functions. Following Buchnall and haundy [3], [4], we define the opeatos A(A) and V(A) as A(A) = "S + h, S 4- h; h, ô + ô' + h whee ô = xd/dx, ô = yd/dy. Now it can be seen without difficulty V(A) = 'h, 5 + h' + h; ô + h,ô' + h Received Febuay 21, 1966. * I am indebted to D. R. P. Agawal fo letting me ead his unpublished manuscipt [1]. t Fo definition and popeties of these functions, see [2]. 590

(2) (3) EXPANSIONS INVOLVING HYPERGEOMETRI FUNTIONS 591 V(a)F A(a)F (4) V(a)A(c)F («) («.); («/) («) (S,); (8/) («m) (8.); (S/) = F ß + 1 I (oy), a v P i(yp) _o- + 1 I (8 ), a; (O, a v + 1 p 4-1 (ft), a; (ft'), a, a <y (O; (8/) Ai 4-1 (am), a v + 1 (0,), e; (ft'), c p 4-1 (tp), c 0-4-1 (8J, a; (ô/), a Then following the method of Buchnall and haundy and using the following identities (5) (6) fh, to + n 4- /i;~] = y \_m + h, n + h } =0 Tm + h,n + h{] [h, m + n + h] y_ %> [l] [ ni][ n] [l] [h) [ m][ n] h m n + l] (Gauss's theoem) (Gauss's theoem with eithe m o n a positive intege) (7) (8) _ y_[ }[h}2[ m][ n] ÍTo [1], [h + - l] [to 4- A] [n + h] (limiting fom of Dougall's theoem) [h,m + n + h to 4- k, n 4- fc m + h,n + h, fc, m 4- n 4- fc ;] [fc A] [fc]2[ m] [ n] -z (9) o [l] [fc + - l] [m + fc], [n 4- fc] [A] (limiting fom of Dougall's theoem) y [h k] [ to], [ n] o [l] [Aj [ fc to n + 1], (Saalschütz's theoem with eithe to o» a positive intege) we get the following five geneal expansions (10) F np. («m) (y,) (SO; (s/) X i7 y [(^)]2[(ft)]Qa][(ft')],, " S[l][W[o]S[(80][(8/)] V ß V + 1 P + l (7 («) + 2 a +, (00 4- ; a +, (0,') + a + 2, + It (80 + ; (ô/) +

592 ARUN VERMA (ID [-n(gp)u(ft)u(ft')] xj/ :U []2, [(SO], t(8.')] [a], («0 4-2, a + (ft) + ; (0/) 4- + 2 (SO +,a [- a], [(«Qk [(ft)], [(ft')] M (12) = t= > [1], [c + - l][]»[a]i[(8.)m(8,')], M + 1 v + 1 P + 1 <+ 1 (-)[(«Qk[(ft)U(ft')], (13) = ^o [1], [a + - 1], [(tow [(8.)] [(8/)], M + 1 c P a + 1 + - («/) 4-, c + a; y («0 + 2, c 4-2 (00 4-, a 4- ; (0/) +, a 4-4- 2, a + 2 (50 4-, c + 2; (S/) +, c + 2 («) 4-2, a + (ft) 4- ; (0/) 4- (yp) 4-2 (50 4-, a + 2; (ô/) +, a + 2 (14) (-)[a c],[(ao]2,[(ft)][(0/)],[a],,, [1], t]2, [a]2, [(50], [(5/)], M + 1 («) 4-2, c + ; V + 1 (ft) +, a + ; (0/) 4-, a 4- P + 1 (to 4-2, a 4-2 ^ <+ 1 (50 4-, c 4- ; (s/) 4-, c + It may be pointed out that the Eq. (10) on specialization of paametes gives Eqs. (26), (29), (32), (35), (36), (38), (41) and ou (11) gives (27), (33), (37), (39); (12) gives (30) and (42); (13) gives Equations (28), (34) and (40); (14) gives Eqs. (31) and (41) of Buchnall and haundy [3]. The above expansions (12) and (13) wee deduced by Ragab [11] by the iteation of seies. 3. In this section, we fist state and then pove some moe geneal expansions F 'A + G (aa), (go) B + H (6B), (A*); (bb'), (A*') Xx (cc) py D + E (dd), (eb); (dd'), (eb) = AA' ^"^m+n^o)]m+n[(ha)]m[(hh)]n[h + am m + 1],-1 (15) m=0 n=0 X [A' 4- ßn - n + i]n-, A B + 2 F E + 2 X(-xTi-yTF [l]m [1]» [(c)un \{dd)]m l(dd')]n (aa) (bb), 1 4- A/a, -m; (6*'), 1 + h'/ß, -n (f) (eb), A 4- «to m 4-1, A/«; (eb), h' + ßn n + I, A/0 F + G 2 + H D (/O + m + n, (ga) + m -\- n (A/0 4- m, A 4- «to; (A«') + n, h + ßn (ce) + m + n (dd) + to; (dd') + n

(16) (17) (18) F A B 1) M B D EXPANSIONS INVOLVING HYPERGEOMETRI FUNTIONS (aa) (bs); (O (e) (dd); (dd') Xx ß!l (-)+í[(«p)uo(ft)],[(ft')]s 0 í=0 [1], [1]. []+S [(8,)] [(8/)] X xvf (ao (60; (6»') (cc) (du); (db') ' P (7 X. M// +P ß + Ul = (-)+,[(«p)uo(ft)],[(ft')]s í=í í=í [1], [1]. []+8 [(80], [(S.')] E + D u + F = x UO, (bb), (50, -; (6B'), (S/), -s (ce), (am) Z) 4- v (d*), (ft); (dß'), (ft') («p) 4-4- s (00 4- ; (0/) + s + + s (8 ) + ; (O 4- s 1 + A + p l + B + a + p D 4- v M ao (,);(,') p +1 a (c*), (dc) (c), (/0; (c/), (/,') (-)+'[(Q]+0(6Q] 5 [l] [1], [c + - 1], [(6 ')]sxy c' + s - 1]. [(e.)],+8 [(/O], [(//)]. w (c,) 4-» + s c 4- + s!],-+» x'y' c + + s 1, (aa), -, (60, (50; s, (O, («/) (cc), (ofm) (dfl), (ft); (dd'), (ft') («0 4-» 4- s (ft) + ; (0/) 4- s c + 2 4-2s, + + s (50 + ; (5/) 4- s X. M.'/ f (60 4-? ;(60 4- s x E (ee) 4-4- s y.1 4- F c + 2, (/O + ; c + 2s, (/,) p (Ap) 2 + q c+ - 1, -, (50; c + s - 1, -s, (B,') w + D (aw), (dd) u + v (cu), (60; (c,/), (6/)

594 ARTJN VERMA To pove (15) equate the coefficients of [(ox)]jf+jy [(bb)]ti [(bb')]n x AT V A p. [1U HI* Ke*)]* [(«-')]* on both the sides and then put M + = to, N 4- s = n afte witing definition fo the Kampé de Féiet's function. We then get 1 = [A 4- am][ti 4- ßN] x the seies (-)+s[(/0 + M + ivwh+.[(ff«) + M 4- N}p+Q++s 0 í=0 5=0 [1]p [1]Q [l] [1]. [(cc) + M + A^]P+e+,+s [(A,) 4- MUP [(Ah') 4- NUq U + A + («- 1) _4- MaUp-i [1 4- A' 4- s(0-1) 4- ATflWi X [(dd) + M],+P [(dd') 4- N]s+Q Next, set -\- P = T, s 4- Q = U and change the ode of summation. Then it is not difficult to see that the coefficient of the tem independent of xy on the ight-hand side is unity. It thus emains to show that the coefficient of x yu on the ight-hand side (T > 0, U > 0) vanish, i.e. n=yy (-)+*[l + A + K«-1)4- Af«]-i [1 + A' 4- s(0-1)4- Nß]y^. í ÚÍ [1], [1], [l]-, [l]uwhich is a known esult due to haundy [6]. It may be emaked that this esult is a genealisation of a esult due to haundy [6] and contains as a special case a esult due to Niblett [10]. To pove (16), equate the coefficients of [(aa)]u+n [(bb)]ul(bb )]n,m n [1]m [1]n 1(cc)}m+n [(do)]* [(dd)u ' M on both the sides and then using the seies definition of the Kampé de Féiet's function and setting 4- m = P and s + n = Q, we get that m n _ []m+at[(5q]m 1(5 )U y y [(«p)]f+o [(ft)]p [(ft')]e p q X V - [(a )W [(ft)]«[(ft')]. í^o ^o []p+0 [(8 )]p [(8,')]c ^ ( P Q I \+s+m+n x/ =0 s-0 [l]_m [l]s-jv [l]p- [1]q Using a lemma due to haundy, we find that îo P = M ; s Q the double seies in the cooked backets is equal to 1 and fo P > s, Q > N its value is zeo and hence the esult is established. This esult incopoates as special case some of the esults due to haundy (Eqs. (8), (9), (10) and (11) of [5]) and fo y = p = 0 this educes to a esult due to Jey L. Fields and Jet Wimp which in tun contains a lage numbe of othe expansions as special cases. To pove (17), equate the coefficient of l(aa)]m+n [(60W l(bb)]s" [l].v [1]* [(cc)w [(do)]m [(dd')w '

EXPANSIONS INVOLVING HYPERGEOMETRI FUNTIONS 595 on both the sides, then use the seies definition of the Kampé de Féiet's function and use the second lemma of haundy [5], afte setting P + = m and Q 4- s = n to get the desied esult. This esult also contains as a special case some of the expansions due to haundy [5] and educes fo y = p = 0 to a esult due to Jey L. Fields and Jet Wimp [7]. The poof of (18) can be developed on exactly simila lines. This esult also gives as special cases some known expansions due to Fields and Wimp. 4. In this section we shall extend the esults deduced in the pevious sections to G-functions of two vaiables. Agawal [1] defined a G-function of two vaiables though a Banes contou integal in the fom p,t,s,q (ao (70; (7/) (so _, -L f f *( +,,)*(, )a;v did* 47T" J so J too whee "(ft,0, (7.O +, (ßm2) V, (7*0 + V, U - (Ä.,+1.,) + 1,1 - (7*1+1.«) -, 1 (0m,+l.O *({+u) = n - (eo +? 4-,; [fn+i.v) f V) (50 4-4- Í?. 77,1 (7.2+l,f) -J' and 0 ^ TOj áí,oáffli í,0á^áí Oáiíáí,0^»áp. The sequence of paametes (0 n), (0m2), (7»0, (7»2) and (e») ae such that none of the poles of the integand coincide. The paths of integation ae indented, if necessay, in such a manne that all the poles of T[ßj Q, j = 1, 2,, Toi, and [0A-' j],k = 1, 2,, m2 lie to the ight and those of [7/4-17], J = 1, 2,, j»2 ; T[yK + ], K = 1, 2,, vi and [l - ey + + ij], j" = 1, 2,, n lie to the left of the imaginay axis. The integal (19) conveges if p + q + s + t < 2(?»! + v! + n), and p + q + s + t < 2(m, + V2 + n), I ag x I < i(mi + vx + n - %[p + q + s + <]), I ag y \ < T(m-2 + v2 + n - %[p + q + s + t\). Now using the technique of Laplace tansfom and its invese as used fo the functions of one vaiable by Luke and Wimp [8] and by the autho fo obtaining geneal expansions of G-functions of Meije [9], one can also wite the extension to (7-functions of all the esults deduced in this pape. Fo instance, one can genealise (10) in the fom

596 ARUN VERMA Gfl.V+t,P+U,V,W tt+ptv+q,p,o+l X!J (ap+o (ft+o;(ftvo 0,(50; 0, (s/)_ («p+p) a,(0,+o; a> (ftv4), a,(ô,);, The poof of this esult is funished by mathematical induction and fo t = u = p = q = 0, v = w = 1 this educes to (10) on using the equation (iii) of 3 and (A) of 4 of Agawal [1]. Similaly all the othe esults can be extended to G-functions of two vaiables. These extended esults wee given ealie by Meije [9], Luke and Wimp [8] and othes. Depatment of Mathematics The Univesity of Goakhpu Goakhpu, India 1-0 [IJ V 1. R. P. Agawal, "An extension of Meije's G-functions," Poc. Nat. Inst. Sei. Sect. A, v. 32, 1965. 2. P. Appell & J. Kampé de Féiet, Fonctions Hype géométiques et Hypespheiques, Gauthie-Villas, Pais, 1926. 3. J. L. Buchnall & T. W. haundy, "Expansions of Appell's double hypegeometic functions," Quat. J. Math. Oxfod Se., v. 11, 1940, pp. 249-270. MR 2, 287. 4. J. L. Buchnall & T. W. haundy, ibid., II, Quat. J. Math. Oxfod Se., v. 12, 1941, pp. 112-128. MR 3, 118. 5. T. W. haundy, "Expansions of hypegeometic functions," Quat. J. Math. Oxfod Se., v. 13, 1942, pp. 159-171. MR 4,197. 6. T. W. haundy, "On lausen's hypegeometic identity," Quat. J. Math. Oxfod Se., 2, v.9, 1958, pp. 265-274. MR 20 #5898. 7. J. L. Fields & J. Wimp, "Expansions of hypegeometic functions in hypegeometic functions," Math. omp., v. 15, 1961, pp. 390-395. MR 23 # A3289. 8. Y. L. Luke & Jet Wimp, "Expansion fomulas fo genealized hypegeometic functions," Rend. ie. Mat. Palemo (2), v. 11, 1962, pp. 351-366. 9.. S. Meije, a) "Expansion theoems fo the G-functions," I, II, Nedel. Akad. Wetensch. Poc. Se. A 55 = Indag. Math., v. 14, 1952, pp. 369-379, 483-487. MR 14, 469; MR 14, 642. b) ibid., Ill, IV, V, Nedel. Akad. Wetensch. Poc. Se. A 56 = Indag. Math., v. 15, 1953, pp. 43-49, 187-193, 349-357. MR 14, 748; MR 14, 979; MR 15, 422. c) ibid., VI, VII, VIII, Nedel. Akad. Wetensch. Poc. Se. A 57 = Indag. Math., v. 16, 1954, pp. 77-82, 83-91, 273-279. MR 15, 791; MR 15, 955. d) ibid., IX, Nedel. Akad. Wetensch. Poc. Se. A 58 = Indag. Math., v. 17, pp. 243-251. MR 16, 1106. 10. J. D. Niblett, "Some hypegeometic identities," Pacifie J. Math., v. 2, 1952, pp. 219-225. MR 13, 940. 11. F. M. Ragab, "Expansions of Kampé de Féiet's double hypegeometic functions of highe ode," J. Reine. An-gew. Math., v. 212, 1963, pp. 113-119. MR 27 #352. 12. A. Vema, "A class of expansions of G-functions and Laplace tansfom," Math. omp., v. 19, 1965, pp. 661-664. (S/)