Automatic generation of hypergeometric identities by the beta integral method

Transcription

1 Journl of Computtionl nd Applid Mthmtics 60 (00) Automtic gnrtion of hyprgomtric idntitis by th bt intgrl mthod C. Krttnthlr ;;, K. Srinivs Ro b; Institut fur Mthmtik dr Univrsitt Win, Strudlhofgss 4, A-090 Win, Austri b Institut of Mthmticl Scincs, Chnni 600, Indi Rcivd 0 Sptmbr 00; rcivd in rvisd form Mrch 00 Abstrct In this rticl, hyprgomtric idntitis (or trnsformtions) for p+ F p -sris nd for Kmp d Frit sris of unit rgumnts r drivd systmticlly from known trnsformtions of hyprgomtric sris nd products of hyprgomtric sris, rspctivly, using th bt intgrl mthod in n utomtd mnnr, bsd on th Mthmtic pckg HYP. As rsult, w obtin som known nd som idntitis which sm to not hv bn rcordd bfor in litrtur. c 00 Elsvir B.V. All rights rsrvd. MSC: primry C0; scondry C05; C70 Kywords: Bt intgrl; Gnrlizd hyprgomtric sris; Kmp d Frit functions. Introduction Eulr s bt intgrl vlution 0 z ( z) dz () () ( + ) ; (.) Corrsponding uthor. E-mil ddrsss: krtt@p.univi.c.t (C. Krttnthlr), ro@imsc.rnt.in (K. Srinivs Ro). URL: krtt Rsrch prtilly supportd by th Austrin Scinc Foundtion FWF, Grnt P094-MAT, nd by EC s IHRP Progrmm, Grnt HPRN-CT Currnt ddrss: Institut Girrd Dsrgus, Univrsit Clud Brnrd Lyon-I,, vnu Clud Brnrd, F-696 Villurbnn Cdx, Frnc /$ - s front mttr c 00 Elsvir B.V. All rights rsrvd. doi:0.06/s (0)0069-0

2 60 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) 59 7 providd R() 0 nd R() 0, is t th hrt of mny idntitis in th thory of hyprgomtric sris. An xmpl is th wll-known intgrl rprsnttion (s,.g.,, (), Sction.6) of hyprgomtric sris ; ;:::; p p+f p ; t ; ;:::; p () ;:::; z ( z) p pf p ; zt dz: (.) () ( ) 0 ;:::; p Hr w us th stndrd hyprgomtric nottion ;:::; r ( ) k ( r ) k rf s ; z z k ; b ;:::;b s k!(b ) k (b s ) k k0 whr th Pochhmmr symbol () k is dnd by () k (+)(+) (+k ), k 0, () 0. (In ordr to prov (.), on would intrchng intgrtion nd summtion on th right-hnd sid, nd thn us (.) to vlut th intgrl insid th summtion.) In this ppr, w propos th xploittion of (wht w cll) th bt intgrl mthod, mthod of driving nw hyprgomtric idntitis from old ons, using th bt intgrl vlution, which is folklor in th hyprgomtric litrtur, lthough pprncs cn b only found spordiclly (s, for xmpl,, Chptr, Exrciss 5, 4 nd 6). It sms tht it ws nvr xploitd systmticlly, probbly bcus of th ort it tks to do ll th computtions. Howvr, with th hlp of computr, ths computtions bcom compltly pinlss. This is wht w propos hr: th compltly utomtic ppliction of th bt intgrl mthod. To convy th id, w briy rcll n rly occurrnc of this mthod (but, vry likly, not th rst) in 8, Sction. W strt with th wll-known trnsformtion formul (4, (.8.0) with ; c rplcd by n; ;, rspctivly) n; F ; z ( ) n () n F n; n + ; z ; (.) multiply both sids by z ( z), intgrt both sids with rspct to z, 06 z 6, intrchng intgrtion nd summtion on both sids, thn us (.), nd nlly convrt th rsult bck to hyprgomtric nottion, to gt th trnsformtion formul: n; ; F ; ( ) n n; ; F ; () n + n; ; ; (.4) In fct, w hv to tmporrily rstrict th prmtrs nd to R() 0 nd R( ) 0, bcus othrwis bt intgrl (.) would not convrg. Howvr, ths rstrictions cn in th nd b rmovd by nlytic continution. W shll, if ncssry, mk similr tmporry ssumptions without mntioning in subsqunt drivtions.

3 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) whr n is nonngtiv intgr. In this cs w obtin n lrdy known trnsformtion formul, nmly on of Thom s F -trnsformtion formuls. (Formul (.4) cn for xmpl b xtrctd from Tbls II A nd II B (Fp(0; 4; 5) Fn(4; 0; )) in Bily s trct 4 which summrizs nd groups th quivlnt numrtor nd dnomintor functions obtind in 7 in th nottion of Whippl 9.) Howvr, not only cn this mthod b pplid to lrg vrity of idntitis, its ppliction cn b compltly utomtizd (s w lrdy nnouncd) with th hlp of th Mthmtic progrm HYP crtd by on of us 9. Thus, this mthod is similr in spirit to th id of dul idntitis in WZ-thory 0 (though it is lss sophistictd), nd th id of prmtr ugmnttion in 6,7 (which, howvr, introducs just on dditionl prmtr instd of two s in th bt intgrl mthod). Also thr, th id is to strt with known idntity, nd thn pply crtin procdur to utomticlly produc (possibly) nwidntity. (In WZ-thory, th procdur consists in nding th WZ-pir which is bhind th originl idntity, nd thn us summtion ovr th othr vribl to obtin dirnt idntity. Prmtr ugmnttion pplis crtin (q-)dirntil oprtor to known idntity.) Th lgorithm hs lso bn pplid to trnsformtion formul for products of hyprgomtric functions known in litrtur. As rsult, formul for th Kmp d Frit sris of unit rgumnts, which trnsform thm into singl-sum hyprgomtric sris, r obtind. Agin, som of th rsults obtind r known ons but som of th rsults obtind sm to b nw nd intrsting. In Sction, th lgorithm nd its implmnttion r prsntd. A sction of th Mthmtic sssion is rproducd to giv fl for th mthodology doptd. In Sction, w list th rsults if w pply our lgorithm to known hyprgomtric trnsformtion formul btwn singl-sum sris. In Sction 4, w list fw of th rsults for Kmp dfrit tht w obtin whn w pply our mchinry to idntitis involving products of hyprgomtric sris. Concluding rmrks r md in Sction 5.. Th lgorithm Lt us rcll th bsic stps of th lgorithm tht producd (.4) from (.): (i) convrt th hyprgomtric sris on both sids of givn trnsformtion into sums; (ii) multiply both sids of th qution by th fctors z ( z) ; (iii) intgrt trm by trm with rspct to z for 0 6 z 6 ; (iv) intrchng intgrtion nd summtion; (v) us th bt intgrl to vlut th intgrls insid th summtions; (vi) convrt th sums bck into hyprgomtric nottion. W giv blow(s In in th Mthmtic sssion blow) n implmnttion of this lgorithm in Mthmtic. Thr, function T is dnd, which is pplid to som qution. To briy xplin th cod: Stp (i) is prformd in th rst lin (XErsEqu,FSUM,{,}), whr th vribl X is st to th qution Equ whr on both sids th hyprgomtric sris is convrtd into sum (which is chivd by th ppliction of FSUM). Thn, in th nxt lin, both sids of th qution r multiplid by z A ( z) B A, thus prforming Stp (ii). Th subsqunt lin

4 6 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) 59 7 hs th purpos of bringing vrything insid th summtions (which is chivd by SUMSmml). Thn, Stps (iii) (vi) r prformd in th following wy. First, th vribl Y is st to th intgrl ovr z, 06 z 6, of th summnd (rprsntd by X,) of th sris on th lft-hnd sid of th qution. Subsquntly, th rsult is summd ovr k, nd thn convrtd to hyprgomtric nottion (th lttr bing chivd by SUMF). Thn th sm is don for th right-hnd sid, nd th rsult is stord in th vribl X. Finlly, in th lst lin (YX) th rsults on both sids r qutd. To illustrt th us of this lgorithm, w djoin sgmnt of th Mthmtic output whr th lgorithm is pplid to th trnsformtion formul 4, (..5). Within HYP, th trnsformtion formul is input s Tgl0 (s In). Thn th lgorithm is invokd, by pplying th function T to th trnsformtion formul. In th intrctiv mod th qustions risd by Mthmtic hv to b nswrd ppropritly. Th rsult is displyd in Out4. Mthmtic. for DOS 87 Copyright Wolfrm Rsrch, Inc. In : hyp:m In :TEqu : Modul{X,Y}, X ErsEqu; FSUM; {; }; Mlz (A ) ( z) (B A ); X (Glichung:SUMSmml); Y IntgrtX; ; {z; 0; }; Y (SUMY; {k; 0; Infinity}:SUMF); X IntgrtX; ; {z; 0; }; X (SUMX; {k; 0; Infinity}:SUMF); Y X In :Tgl0 Do you wnt to st vlus for th qution? y n: n b + c + c; b + c Out F c ; z ( z) F ; z c In4 :T% Is - nonngtiv intgr? y n: n Is -b nonngtiv intgr? y n: n A hyprgomtric sris is convrtd into sum. Entr vribl for th summtion indx: k Is -c nonngtiv intgr?

5 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) y n: n Is b-c nonngtiv intgr? y n: n A hyprgomtric sris is convrtd into sum. Entr vribl for th summtion indx: k A; F B; c ; (A) ( A + B) Out4 A; + c; b + c b + B + c; c ; (B) (A) ( A b + B + c) F ( b + B + c) Notic tht (A) is occurring on both sids of th output qution. Simpliction of th rsult cn b don ithr by hnd or using th vrious commnds vilbl in HYP. Aftr th following rplcmnts, in succssion: c, A c, B d, th nl rsult obtind is c; + ; b + F ; c d; ; (d) (s) ( c + d) (s + c) F s + c; d ; ; (.) whr s d + b c, is th prmtr xcss. This idntity blongs to th st of 0 nontrminting F sris (s 4, Exmpl 7, p. 98 nd 5, (III)). Within HYP, this output cn thn b convrtd into ny of th commonly usd forms of TEX (LATEX, AMS-LATEX, AMS- TEX or Plin-TEX). In th cs of this rticl, convrsion into AMS-LATEX cod ws pplid, which is rproducd hr.. Nw singl sum hyprgomtric idntitis from old ons In this sction, w pply th lgorithm of th prvious sction systmticlly to known hyprgomtric sris trnsformtions. Whnvr w hv bn bl to trc th obtind idntity in th litrtur, w provid th rspctiv rfrnc. In ny cs, w hv lrdy sn two such xmpls: If w strt with trnsformtion formul (.), thn th lgorithm rsults in (.4), nd if w strt with th trnsformtion formul displyd s Out of th Mthmtic sssion, thn th lgorithm rsults in (.). Th trnsformtion formul 4, (.7..) F c ; z ; c b ( z) F c ; z z will lso rsult in on of th 8 trminting F trnsformtions (s 6, (IX), p. 9), whn on of th numrtor prmtrs is ngtiv intgr.

6 64 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) If w strt with th qudrtic trnsformtion formul, (.) F + b ; z ( z) F ; + + b ; 4z ( z) nd ssum tht is nonpositiv intgr (so tht th F -sris trmint), thn w obtin ; d F + b; ; () ( d) ( ) ( d) 4 F + ; ;d;+ + b; + + d ; + + d (.) ; : (.) This trnsformtion rlting nrly-poisd F to 4 F is vlid providd is nonpositiv intgr. Howvr, by stndrd polynomil trick, it cn b sn tht it is lso tru if is rbitrry but d is nonpositiv intgr: lt d by xd nonpositiv intgr. By multiplying both sids of (.) by ( ) d ( + b) d ( + + d ) d ; both sids bcom polynomils in of dgr t most 5d. Ths two polynomils gr for ll nonpositiv, sinc w know tht (.) is tru for nonpositiv. Ths r innitly mny vlus of, whnc th polynomils must b idnticl.. If w strt with th trnsformtion formul, (5.0) ; F + + b ; z ( z) F b ; z z nd ssum tht is nonpositiv intgr, thn w obtin ; 4F + ; + d ; d () ( d) + b; + ; ; ( ) ( d) F ; d b; + + d ; (.) : (.4) This idntity is tru providd is nonpositiv intgr or d is nonpositiv intgr, th lttr bcus of th sm rgumnts s sktchd in itm. It cn b found in th litrtur s spcil cs of mor gnrl trnsformtion for bsic hyprgomtric sris (s 8, Exrcis.4, q, rvrsd; it is vilbl s T5 within HYP, s 0).. If w strt with th qudrtic trnsformtion formul of Gu (4, Exmpl 4.(iii), p. 97, with, b, x z) F + + b ; z F ; b + + b ; 4( z)z nd ssum tht is nonpositiv intgr, thn w obtin ; d F + + b ;; 4 F ; b ;d; d b ; + ; ; : (.6) (.5)

7 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) This idntity is tru providd both hyprgomtric sris trmint. It is th min thorm in (ctully, thr vn q-nlogu of (.6) cn b found; s lso 8, (.0.); Appndix (III.)). This sm rsult (.6) will b obtind whn z z in (.5). 4. If w strt with th qudrtic trnsformtion formul, (5.), rvrsd F b ; z ( z) F ; + b + b ; z 4(z ) (.7) nd ssum tht is n vn nonpositiv intgr, thn w obtin ; d F b; ; () ( d) ( ) ( d) 4 F ; + b; + d ; d + b; + + d ; + ; ; (.8) providd is n vn nonpositiv intgr or d is ny nonpositiv intgr. This idntity cn lso b obtind in dirnt (but mor complictd) wy: In th trnsformtion formul listd s T49 in HYP (s 0; it is Eq. (.5.7) from 8 with q ) lt. Thn on th right-hnd sid th scond trm vnishs, whil th rst is vry-wll-poisd 7 F 6 -sris to which Whippl s 7 F 6 -to- 4 F trnsformtion (s 4, (.4..)) cn b pplid. Th rsult is trnsformtion formul (.8). 5. Th trnsformtion formul, (.), rvrsd ; F ; z ( z) c F + c ; + + c ;4( z) z c c (.9) rsults in ; ; d F ; c; () (c d + ) (c + ) ( d) 4 F + c ; + + c ;d;c d + c; + c + ; c + ; ; (.0) providd both hyprgomtric sris trmint. 6. If w strt with th trnsformtion formul (, (4.0), with x z) F b ; 4 z ( z) ( z) F ; + b + b ; z ; (.)

8 66 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) 59 7 which is combintion of, (5.0) nd, (6.), rvrsd, nd ssum tht is nonpositiv intgr, thn w obtin ; d; 4F b; d + ; + d + ; ( ) (+ d ) ; (+ ) ( d ) 4 F + b; + d ; d + b; + ; + ; ; (.) providd or d is nonpositiv intgr. 7. If w strt with th trnsformtion formul (, (4.), with, b, x z nd, (5.), with z z( z)): F + + b ; z 4(z ) ( z) F ; + b +b ; z (.) nd ssum tht is nonpositiv intgr, thn w obtin ; 4F + d ; d () ( d + ) ; + b; d ; + + b; +d ; ( + ) ( d) F +b; + ; ; (.4) providd or d is nonpositiv intgr. 8. Th trnsformtion formul (, (.) with z 4z( z), + c +, c b) F ;4( z) z ( z) b F + b; + b + + b + + b ; z (.5) nd ssum tht is nonpositiv intgr, thn w obtin ; d; d 4F + + b; + ; ; () ( b d + ) ( b + ) ( d) F + b; + b; d + + b; b + ; ; (.6) providd both hyprgomtric sris trmint. This trnsformtion cn lso b obtind by combining th 4 F -to- F trnsformtion from tht occurrd lrdy in itm with F -trnsformtion (.). 9. If w strt with th trnsformtion formul (8, (.4.8) q, rvrsd) ; ( + z) F + ; 4 z ; ( z) F + ; b b ( z) + ; z (.7)

9 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) nd ssum tht is nonpositiv intgr, thn w obtin ; + ; d; (d + ) 4 F b; + d ; + d ; ( + ) ( d + ) ; + ; b; d 4F ; ; (.8) ( + ) ( d) + ; + providd or d is nonpositiv intgr. (In fct, whn pplying th bt intgrl mthod, w hv to dl with sum of two sris on th lft-hnd sid, which gnrts th fctor (d + ) inth rsult.) 0. If w strt with th trnsformtion formul (4, p. 97, Exmpl 4(iv) with x z) F ; + ; + b c ; 4 z + b; + c ( z) ( z) F ; c + b; + c ; z (.9) nd ssum tht is nonpositiv intgr, thn w obtin 5F + ; ; + b c; d; 4 + b; + c; + d ; + d ; () ( d + ) ; c; d ( + ) ( d) 4 F + b; + c; + ; ; (.0) providd or d is nonpositiv intgr. This is known trnsformtion btwn nrly-poisd 4F sris nd Slschutzin 5 F 4 sris (4, (.4..); to s this do th rplcmnts f, b +f h, c h, g f, in(.0)).. If w strt with th trnsformtion formul (4, p. 97, Exmpl 6 with b + b, c + c, x z) ( + z) F + ; + ; + b + c ; 4 z b; c (z ) ; + ; + b; + c ( z) + 4F ;b;c ; z nd ssum tht is nonpositiv intgr, thn w obtin 5F + ; + ; + b + c ;d; 4 ; d + b; c; + d ; + d ( + ) ( + d + ) ( + + ) ( d) (.) ; + ; + b; + c; d 5F 4 ;b;c;+ + ; ; (.)

10 68 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) 59 7 providd or d is nonpositiv intgr. On rplcing b + b; c + c; d m; + w, this idntity corrsponds to 4, (4.5.). (Th prnthticl rmrk in itm 9 pplis lso hr. Howvr, in th rsult, w movd th fctor (d + ), which is gnrtd by th sum of two sris on th lft-hnd sid of (.), to th right-hnd sid.). If w strt with th trnsformtion formul, (5.8) ( z ) +; b; +b; 4 4F + + b; + b; + ; z 4( z) +; +b; ( z)( z) + 4F + b; 5 + +b; + b; + ;4( z) z (.) nd ssum tht is ngtiv intgr, thn w obtin 4 6F + ; +; b; +b; + d ; d 5 + ; + b; ; + b; +d ; + (+ d + ) ( + ) ( + d + ) d ( + + ) ( d) 5 6 F + ; +; + b; +b; d; + d ; + b; +b; + + ; + + ; ; (.4) providd is ngtiv intgr or d is nonpositiv intgr. (Agin, th prnthticl rmrk of itm 9 pplis, this tim on both sids. Th fctors gnrtd ppr in th rst trm on th right-hnd sid.). If w strt with th cubic trnsformtion formul (, (4.05), with b, b +, x z) ; ( z) F + ; + 7 z ; b; + b 4( z) F ; +b ; + b b; + b ; z 4 (.5) nd ssum tht is nonpositiv intgr, thn w obtin ; 6F + ; + ; d; + ; b; b; + + d ; + + d ; + + d ; ( + ) ( d) ; + b; +b ;d () ( d + ) 4 F + b; b; ; ; (.6) 4

11 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) providd or d is nonpositiv intgr. This is n unusul idntity fturing trnsformtion btwn 6 F 5 -sris nd 4 F 4 -sris. 4. If w strt with th scond cubic trnsformtion formul of Bily (4, (4.06) with b, + b, x z): ; F + ; + b; + b ; 7 z 4( z) ; + b; + b ( z) F b ; +6 b ;4z (.7) nd ssum tht is nonpositiv intgr, thn w obtin ; 6F + ; + ; + d ; d ; 5 + b; b; d + ; + d + ; + d + ; ( ) (+ d ) ; + b; (+ ) ( d ) 4 F + b; d +6 b; b ; + ;4 ; (.8) providd or d is nonpositiv intgr. 5. If w strt with th trnsformtion formul 5, Entry 4 of Rmnujn, Chptr, p. 50: F ; + + b ; 4 z ( z) ( z) F ; + b nd ssum tht is nonpositiv intgr, thn w obtin 4F + ; ;d; + b; d + ; + d + ; ( ) ( + d ) ( + ) ( d ) F ; + b; d + b ; z ; + b; + (.9) ; (.0) providd or d is nonpositiv intgr. 6. Th trnsformtion formul (4, (.8.0), with c rplcd by + + b c nd z rplcd by z) xprssing th Gu solution vlid for z, in trms of th Gu functions vlid for z is F c ; z ( z) + c b (c) ( + b c) () (b) (c) (c b) (c b) (c ) F b + c; + c F b + c ; z + + b c ; z : (.)

12 70 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) 59 7 If our lgorithm is pplid, it rsults in ; c F d; ; (d) (d b) (d ) (d b) F ; c b d; ; ( + b d) (d) () (d + b c) + () (b) ( c) (d + b) d ; d b; d + b c F +d b; d + b ; (.) This thr-trm F -trnsformtion formul cn for xmpl b found in 4, (4..4.). 7. Th trnsformtion formul 5, Entry of Rmnujn F + + ; z z ( ) ( + +b) F + ; + b b ; z () (b) rsults in + ( ) ( + +b) ( + ) ( + b) F F ; d + + b ;; ; b ; z ( ) ( + +b) ( + ) ( + b) 4 F ; b ; + d ; d ; + ; ; + ( ) ( + +b) ( + d) () 4F + ; + b ; + d ; + d () (b) (d) ( + ) ; + ; + (.) ; : (.4) 4. Products of hyprgomtric sris nd idntitis for Kmp d Frit sris Thr xist mny rltions conncting products of hyprgomtric sris, including th ons clld th thorms of Cyly nd Orr 4, Sction 0.. In this sction, w pply our mchinry to idntitis of th form product of two hyprgomtric sris hyprgomtric sris: (4.) As bfor, on th right-hnd sid w will obtin nw hyprgomtric sris. Howvr, w will s tht on th lft-hnd sid w obtin Kmp dfrit sris. Ths r hyprgomtric doubl sris. Th stndrd nottion, which will b usd in th squl, for ths is s follows: () : (b) ; (b ) ; F A:B;B C:D;D (c) : (d) ; (d z ;z A j ( B j) m+n j (b B j) m j (b j) n z m C ) ; j (c D j) m+n j (d zn D j) m j (d j ) n m! n! : m;n 0

13 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) W rport blowth intrsting rsults w obtin from th idntitis of form (4.) found in 4, Sction.5.. Th stndrd thorm of th typ (4.) is th clbrtd Clusn s thorm. It concrns th squr of Gu sris nd it is F + + b ; z ; + b F +b; + + b ; z : (4.) Th Mthmtic progrm givn in Sction bov hs bn modid to tk cr of css lik this nd it gnrtd th following rsult from (4.): d : ; ; ; + b; d F :; :; : + + b; + + b; x; x 4 F ; x : (4.) + + b; +b; (Th rdr should not tht th bov rsult is obtind by rst rplcing z by xz in (4.) nd subsquntly pplying our mthod.). Th trnsformtion which follows from Thorm VII of Bily s 4, (.5.) F + + b c ; z F c ; z ; 4 F + b ; + + b ;4( z) z + b; c; + + b c rsults in d : ; ; F :; :; ; : + + b c; c; ; 6 F + b ; + + b ;b;d; d 5 + b; + + b c; c; + ; ; : (4.5) (4.4). If w strt with th trnsformtion which follows from Thorm VIII of 4, (.5.) ; b + c ; + c; b + c F c ; z F ; z ( z) 4F c c; c ; + ; z c 4( z) (4.6) nd ssum tht or b is nonpositiv intgr, thn w obtin d : ; ; b + c; F :; :; ; : c; c; ; + c; b + c; + d ; d () ( d) ( ) ( d) 6 F 5 providd, b, or d is nonpositiv intgr. + c ; ; c ;c;+ + d ; + ; (4.7)

14 7 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) Concluding rmrks W hv considrd in this rticl mthod, which w clld th bt intgrl mthod, to obtin trnsformtions nd summtion thorms from known trnsformtions of hyprgomtric nd products of hyprgomtric sris. Whil bing itslf folklor, this mthod ws utomtd with th hlp of progrms writtn in Mthmtic using th softwr pckg HYP dvlopd by on of us 9. W obtind som known idntitis (which w displyd only prtilly), nd som which w wr not bl to trc in th litrtur, mong thm som vry intrsting ons, such s (.0), (.4) (.6), or (.8). Clrly, vritions (tht cn b gin utomtd) my lso b considrd. For xmpl, instd of using th bt intgrl vlution, w my rplc z by zt t th bginning nd thn us th Eulr intgrl rprsnttion (.) (with p )of F t-sris. This yilds idntitis which qut singl sums to sums of F -sris (sid from som xplicit multiplictiv fctors). Anothr dirction would b to us othr xtnsions of th bt intgrl vlution, most notbly q-nlogus throf. Howvr, it sms tht, in prticulr in th cs of q-nlogus, th rprtory of idntitis which cn b usd s strting point, is vry limitd. Rfrncs G.E. Andrws, R. Asky, R. Roy, Spcil functions, Encyclopdi of Mthmtics nd its Applictions, Vol. 7, Cmbridg Univrsity Prss, Cmbridg, 999. G.E. Andrws, D. Stnton, Dtrminnts in pln prtition numrtion, Europ. J. Combin. 9 (998) 7 8. W.N. Bily, Products of gnrlizd hyprgomtric sris, Proc. London Mth. Soc. () 8 (98) W.N. Bily, Gnrlizd Hyprgomtric Sris, Cmbridg Univrsity Prss, Cmbridg, Bruc C. Brndt, Rmnujn s Notbooks, Prt III, Springr, Brlin, W.Y.C. Chn, Z.-G. Liu, Prmtr ugmnttion for bsic hyprgomtric sris, II, J. Combin. Thory Sr. A 80 (997) W.Y.C. Chn, Z.-G. Liu, Prmtr ugmnttion for bsic hyprgomtric sris, I, in: B.E. Sgn, R.P. Stnly (Eds.), Mthmticl Essys in Honor of Gin-Crlo Rot, Progrss in Mthmtics, Vol. 6, Birkhusr, Boston, 998, pp G. Gspr, M. Rhmn, Bsic hyprgomtric sris, Encyclopdi of Mthmtics nd its Applictions, Vol. 5, Cmbridg Univrsity Prss, Cmbridg, C. Krttnthlr, HYP nd HYPQ Mthmtic pckgs for th mnipultion of binomil sums nd hyprgomtric sris rspctivly q-binomil sums nd bsic hyprgomtric sris, J. Symbol. Comput. 0 (995) C. Krttnthlr, HYP, Mnul for Mthmtic pckg for hndling hyprgomtric sris. Avilbl from ( krtt). Y.L. Luk, Th Spcil Functions nd thir Approximtions, Vol. I, Acdmic Prss, London, 969. M. Rhmn, A. Vrm, Qudrtic trnsformtion formuls for bsic hyprgomtric sris, Trns. Amr. Mth. Soc. 5 (99) V.N. Singh, Th bsic nlogus of idntitis of th Cyly Orr typ, J. London Mth. Soc. 4 (959) 5. 4 L.J. Sltr, Gnrlizd Hyprgomtric Functions, Cmbridg Univrsity Prss, Cmbridg, K. Srinivs Ro, H.-D. Dobnr, P. Nttrmnn, Group Thorticl bsis for som trnsformtions of gnrlizd hyprgomtric sris nd th symmtris of th -j nd 6-j cocints, in: P. Ksprkovitz, D. Gru (Eds.), Procdings of th Fifth Wignr Symposium, World Scintic, Singpor, 998, pp K. Srinivs Ro, V. Rjswri, Quntum Thory of Angulr Momntum: Slctd Topics, Nros Publishing Hous, Springr, Brlin, 99.

15 C. Krttnthlr, K. Srinivs Ro / Journl of Computtionl nd Applid Mthmtics 60 (00) J. Thom, Ubr di Funktionn, wlch durch Rihn von dr Form drgstllt wrdn: :::, J. Rin Angw. Mth. 87 (879) M. Wbr, A. Erdlyi, On th nit dirnc nlogu of Rodrigus formul, Amr. Mth. Monthly 59 (95) F.J.W. Whippl, A group of gnrlizd hyprgomtric sris: rltions btwn 0 llid sris of th typ F; c; d;, Proc. London Mth. Soc. () (95) H.S. Wilf, D. Zilbrgr, Rtionl functions tht crtify combintoril idntitis, J. Amr. Mth. Soc. (990)