HyergeometricPFQ Notatios Traditioal ame Geeralied hyergeometric fuctio Traditioal otatio F a 1,, a ; b 1,, b ; Mathematica StadardForm otatio HyergeometricPFQa 1,, a, b 1,, b, Primary defiitio 07.31.0.0001.01 F a 1,, a ; b 1,, b ; a j k k b j k k 1 1 1 1 Re b j a j 0 ; 1 I the cases 1 the series above does ot coverge but it (together with symbol) ca be used as asymtotic series, where, whe eeded a Borel summatio is imlicitly uderstood. 07.31.0.000.01 F a 1,, a ; b 1,, b ; a j k k b j k k ; a j a j For a i, b j m ; m beig oositive itegers ad ak a k a k bk b k m b k the fuctio F a 1,, a ; b 1,, b ; caot be uiuely defied by a limitig rocedure based o the above defiitio because the two variables a i, b j ca aroach oositive itegers, m; m at differet seeds. For the above coditios we defie: 07.31.0.0003.01 F a 1,, a i,, a ; b 1,, b j,, b ; a j k k b j k k ; a i b j m m m
htt://fuctios.wolfram.com Geeral characteristics Some abbreviatios 07.31.04.0001.01 a 1,, a a 1 a Domai ad aalyticity F a 1,, a ; b 1,, b ; is a aalytical fuctio of a 1,, a, b 1,, b ad which is defied i 1. I the cases for fixed a 1,, a, b 1,, b, it is a etire fuctio of. If arameters a k iclude egative itegers, the fuctio F a 1,, a ; b 1,, b ; degeerates to a olyomial i. 07.31.04.000.01 a 1 a b 1 b F a 1,, a ; b 1,, b ; Symmetries ad eriodicities Mirror symmetry 07.31.04.0003.0 F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; ; 1, 1 Permutatio symmetry 07.31.04.0004.01 F a 1, a,, a k,, a j,, a ; b 1,, b ; F a 1, a,, a j,, a k,, a ; b 1,, b ; ; a k a j k j 07.31.04.0005.01 F a 1,, a ; b 1, b,, b k,, b j,, b ; F a 1,, a ; b 1, b,, b j,, b k,, b ; ; b k b j k j Periodicity No eriodicity Poles ad essetial sigularities With resect to For 1 ad fixed a l, b j i oolyomial cases (whe a 1 a ), the fuctio F a 1,, a ; b 1,, b ; does ot have oles ad essetial sigularities. 07.31.04.0006.01 ig F a 1,, a ; b 1,, b ; ; 1 a 1,, a For ad fixed a l, b j i oolyomial cases (whe a 1 a ), the fuctio F a 1,, a ; b 1,, b ; has oly oe sigular oit at. It is a essetial sigular oit. 07.31.04.0007.01 ig F a 1,, a ; b 1,, b ;, ; a 1,, a
htt://fuctios.wolfram.com 3 If arameters a k iclude r egative itegers Α k, the fuctio F a 1,, a ; b 1,, b ; is a olyomial ad has ole of order miα 1,, Α r at. 07.31.04.0008.01 ig F a 1,, a ; b 1,, b ;, Α ; a 1,, a Α mia s1,, a sr a sk With resect to a l The fuctio F a 1,, a ; b 1,, b ; as a fuctio of a l, 1 l, has oly oe sigular oit at a l. It is a essetial sigular oit. 07.31.04.0009.01 ig al F a 1,, a ; b 1,, b ;, ; 1 l With resect to b j The fuctio F a 1,, a ; b 1,, b ; as a fuctio of b j, 1 j, has a ifiite set of sigular oits: a) b j k ; k, are the simle oles with residues 1k k F a 1,, a ; b 1,, b j1, k, b j1,, b ; ; b) b j is the oit of accumulatio of oles, which is a essetial sigular oit. 07.31.04.0010.01 ig b j F a 1,, a ; b 1,, b ; k, 1 ; k,, ; 1 j 07.31.04.0011.01 res b j F a 1,, a ; b 1,, b ; k 1k F a 1,, a ; b 1,, b j1, k, b j1,, b ; ; k 1 j k Brach oits With resect to The fuctio F a 1,, a ; b 1,, b ; does ot have brach oits for ad has two brach oits: 1, for 1 i oolyomial case (whe a 1 a ) 07.31.04.001.01 F a 1,, a ; b 1,, b ; ; 07.31.04.0013.01 1 F a 1,, a 1 ; b 1,, b ; 1, ; a 1,, a 1 07.31.04.0014.01 1 1 F a 1,, a 1 ; b 1,, b ;, 1 log ; Ψ Ψ Ψ b j a j a 1,, a 1 07.31.04.0015.01 1 F a 1,, a 1 ; b 1,, b ;, 1 s ; 1 Ψ b j a j r s r s 1 gcdr, s 1 a 1,, a 1
htt://fuctios.wolfram.com 4 07.31.04.0016.01 1 F a 1,, a 1 ; b 1,, b ;, log ; ai,a j a i a j 1 i 1 1 j 1 i j a 1 a 1 07.31.04.0017.01 1 F a 1,, a 1 ; b 1,, b ;, lcms 1,, s 1 ; a l r l r l, s l s l 1 gcdr l, s l 1 1 l 1 a 1,, a 1 s l With resect to a l The fuctio F a 1,, a ; b 1,, b ; as a fuctio of a l, 1 l, does ot have brach oits. 07.31.04.0018.01 al F a 1,, a ; b 1,, b ; ; 1 l With resect to b j The fuctio F a 1,, a ; b 1,, b ; as a fuctio of b j, 1 j, does ot have brach oits. 07.31.04.0019.01 b j F a 1,, a ; b 1,, b ; ; 1 j Brach cuts With resect to For all oegative iteger a k, the fuctio F a 1,, a ; b 1,, b ; i the cases 1 is a sigle-valued fuctio o the -lae cut alog the iterval 1,, where it is cotiuous from below. I the cases this fuctio does ot have brach cuts. 07.31.04.000.01 1 F a 1,, a 1 ; b 1,, b ; 1,, ; a 1,, a 1 07.31.04.001.01 F a 1,, a ; b 1,, b ; ; 07.31.04.00.01 lim Ε0 1 F a 1,, a 1 ; b 1,, b ; x Ε 1 F a 1,, a 1 ; b 1,, b ; x ; x 1 07.31.04.006.01 lim Ε0 1 F a 1,, a 1 ; b 1,, b ; x Ε b k 1 ak 1,1 G 1,1 Π 1 x 1, b 1,, b a 1,, a 1 ; x 1
htt://fuctios.wolfram.com 5 07.31.04.003.01 lim Ε0 1 F a 1,, a 1 ; b 1,, b ; x Ε b k 1 ak 1 a k b j a k 1 a j a k Π a kx a k 1F a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 x ; j,k,j,k 1 j 11 k 1 a j a k x 1 With resect to a l The fuctio F a 1,, a ; b 1,, b ; as a fuctio of a l, 1 l, does ot have brach cuts. 07.31.04.004.01 al F a 1,, a ; b 1,, b ; ; 1 l With resect to b j The fuctio F a 1,, a ; b 1,, b ; as a fuctio of b j, 1 j, does ot have brach cuts. 07.31.04.005.01 b j F a 1,, a ; b 1,, b ; ; 1 j Series reresetatios Geeralied ower series Exasios at geeric oit 0 For the fuctio itself 07.31.06.0045.01 k 0 F a 1,, a ; b 1,, b ; b j k 1F 1 1, a 1,, a ; 1 k, b 1,, b ; 0 0 k ; 1 1 0 1, 07.31.06.0046.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 1 a j a k b j a k 1 0 a k arg 0 Π 0 a k arg 0 1 Π a k j 0 j j j 0 1F a k b 1 1,, a k b 1, j a k ; 1a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 0 0 j ; 0 1 j,k,j,k 1 j 11 k 1 a j a k Exasios o brach cuts for 1
htt://fuctios.wolfram.com 6 For the fuctio itself 07.31.06.0047.01 1F a 1,, a 1 ; b 1,, b ; b k 1 k G 1,1 1,1 1 ak x argx Π Π 1 a 1 k,, 1 a 1 k 0, 1 b 1 k,, 1 b k xk ; x 1 07.31.06.0048.01 1F a 1,, a 1 ; b 1,, b ; 07.31.06.0049.01 b k 1 ak 1 k G 1,1 1,1 argx Π 1 Π 1 x 1, k b 1,, k b k a 1,, k a 1 x k ; x 1 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 1 x u u a k a j a k Π a k u 0 u b j a k argx 1Floor Π x a k 1F u a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 x x u ; j,k,j,k 1 j 11 k 1 a j a k x 1 Exasios at 0 07.31.06.0001.01 F a 1,, a ; b 1,, b ; 1 07.31.06.000.01 F a 1,, a ; b 1,, b ; a j b j a j k k b j k k a j a j 1 b j b j 1 ; 1 1 ; 1 1 07.31.06.0003.01 F a 1,, a ; b 1,, b ; 1 O Exasios at 1 for 1 The oit 1 is the ed oit of the brach cut for the fuctio 1 F a 1,, a 1 ; b 1,, b ;, where it has a 1 rather comlicated behavior. The corresodig geeral formula (for oiteger Ψ b j aj ) icludes two major terms - regular ad sigular which are a aalytical fuctios. Moreover, the sigular term has reresetatio of the form cost 1 Ψ 1 O 1 ad regular term is bouded ear oit 1. A more detailed descritio of this behavior is reseted below. At the sigular oit 1 the fuctio 1 F a 1,, a 1 ; b 1,, b ; is cotiuous for ReΨ 0, bouded for ReΨ 0, Ψ 0 ad has, i geeral, a logarithmic sigularity for Ψ 0 while for ReΨ 0 it has a ower sigularity of order Ψ to which for iteger Ψ a logarithmic sigularity ca also occur. The geeral formulas
htt://fuctios.wolfram.com 7 07.31.06.0004.01 1F a 1,, a 1 ; b 1,, b ; b j F a 1,, a 1 ;, 1, b 1,, b ; 07.31.06.0005.01 1F a 1,, a 1 ; b 1,, b ; b k 1 Ψ g k Ψ 1 k g k 0 1 k 1 ak ; 1 k k r a 1 k r a Ψ r k 1 1 1 g k r k Ψ r k j j a1,, a 1, b 1,, b 1 k Ψ r k j 0 j a 1 Ψ j a Ψ k k r b j 1 k r a j 1 lim m Ψ r k 1F k r a 1, k r a,, r a 1, m; k r b 1, k r b,, r b, k m r Ψ 1; 1 g 0 Ψ Ψ k1 a1,, a 1, b 1,, b b 1 a 3 1 k1 b j k 1 j j 3 a j k 1 k 1 k 0 k 1 k 1 j b j j 3 a j a 3 b 1 k 1 1 k k b a 4 1 k b j k j 3 j 4 a j k k k 3 0 k k3 1 j 3 b j j 4 a j a 4 b k 1 k3 k3 b a 1 k b j k j a j k k k 1 0 k k1 b a 1 k1 b 1 a 1 k1 1 b j j a j a b k 1 k k1 k1 1 Ψ b j a j Ψ Rea 3 0 Rea 1 0 The logarithmic cases
htt://fuctios.wolfram.com 8 07.31.06.0006.01 1F a 1,, a 1 ; b 1,, b ; b k Ψ 1 k j 1 j j 1 Ψ j j log1 1 j 1 ak j 0 j 0 ; 1 1 Ψ b j a j 1 1 k j 1 j j a 1 j a j k j Ψ 1 k a1,, a 1, b 1,, b ; Rea 3 j k a 1 Ψ k a Ψ 1 Ψ a 1 Ψ j a Ψ j k j 1 Rea 1 j j 1 j k a1,, a 1, b 1,, b j j j Ψ k j1 a 1 Ψ k a Ψ k j j k k a1,, a 1, b 1,, b Ψj k 1 Ψj Ψ 1 Ψj a 1 Ψ Ψj a Ψ ; a 1 Ψ k a Ψ k Rea 3 j Ψ Rea 1 j Ψ 1 Ψ1 a 1 Ψ j a Ψ j j j j Ψ j j k k a1,, a 1, b 1,, b Ψ a 1 Ψ k a Ψ k 07.31.06.0007.01 1F a 1,, a 1 ; b 1,, b ; b k a 1 Ψ j a Ψ j 1 j j k j 1 1 Ψ k a1,, a 1, b 1,, b j j Ψ a 1 Ψ k a Ψ k 1 ak j 0 k j1 j j k log1 Ψj k 1 Ψj Ψ 1 Ψj a 1 Ψ Ψj a Ψ a 1 Ψ k a Ψ k k a1,, a 1, b 1,, b 1 j Ψ 1 j a 1 j a k j Ψ 1 k a1,, a 1, b 1,, b 1 j j k a 1 Ψ k a Ψ j 0 ; 1 1 1 Ψ b j a j 1 Ψ
htt://fuctios.wolfram.com 9 07.31.06.0008.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak Ψ 1 1 Ψ h j 1 j j u j v j log1 1 j j 0 j 0 ; 1 1 1 1 j a 1 Ψ j a Ψ j j Ψ 1 j j k k a1,, a 1, b 1,, b Ψ b j a j 1 h j j a 1 Ψ k a Ψ k a 1 Ψ jψ a Ψ jψ u j 1 j j Ψ j j Ψ k jψ 1 k j Ψ 1 k a1,, a 1, b 1,, b a 1 Ψ k a Ψ k jψ Ψ j 1 Ψ k k a1,, a 1, b 1,, b Ψj 1 Ψj a 1 Ψj a Ψj k Ψ 1 ; a 1 Ψ k a Ψ k 1 Ψ1 a 1 Ψ jψ a Ψ jψ Rea 3 j Rea 1 j v j j j Ψ jψ Ψ j k k a1,, a 1, b 1,, b Ψ a 1 Ψ k a Ψ k 07.31.06.0009.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak Ψ 1 j 1 k 1 Ψ k a1,, a 1, b 1,, b 1 j j Ψ 1 a 1 Ψ j a Ψ j j 0 j k a 1 Ψ k a Ψ k 1 j k a 1 Ψ 1 j k a k a1,, a 1, b 1,, b j Ψ j k k a 1 Ψ k a Ψ j Ψ log1 Ψj k 1 Ψj Ψ 1 Ψj k a 1 Ψj k a 1 j Ψ 1 j k a 1 Ψ1 1 j k a k a1,, a 1, b 1,, b 1 j j k k a 1 Ψ k a Ψ j k ; 1 1 1 Ψ b j a j 1 Ψ
htt://fuctios.wolfram.com 10 07.31.06.0010.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak a 1 j a j j j k log1 Ψj 1 Ψj a 1 Ψj a Ψj k 1 k a1,, a 1, b 1,, b j 0 j a 1 k a k 1 j j k j1 k j 1 a 1 k a k 1 1 Ψ b j a j 1 Ψ 0 1 k a1,, a 1, b 1,, b 1 j ; The major terms i the geeral formula for exasios of fuctio 1F a 1,, a 1 ; b 1,, b ; at 1 07.31.06.0011.01 1F a 1,, a 1 ; b 1,, b ; 1 F a 1,, a 1 ; b 1,, b ; 11 O 1 1 Ψ b j a j Ψ 07.31.06.001.01 1 1F a 1,, a 1 ; b 1,, b ; Ψ 1 k 3 ak b k Ψ k k a1,, a 1, b 1,, b 1 O 1 k a 1 Ψ k a Ψ Ψ Ψ 1 ak 1 k 3 ak 1 Ψ b j a j 1 ReΨ 0 Rea 3 0 Rea 1 0 07.31.06.0013.01 1 1F a 1,, a 1 ; b 1,, b ; 1 F a 1,, a 1 ; b 1,, b ; 11 O 1 1 Ψ b j a j Ψ 0 1 Exasios at for 1 b j 1 a j b k 1 Ψ 1 O 1 ; b k 1 Ψ 1 O 1 ; log1 1 O 1 ; The geeral formulas 07.31.06.0014.01 1F a 1,, a 1 ; b 1,, b ; b k F a 1,, a ; b 1,, b ;,, ; 0, 1
htt://fuctios.wolfram.com 11 07.31.06.0015.01 1F a 1,, a 1 ; b 1,, b ; b k ower a 1,, a ; F b 1,, b ;,, ; 0, 1 Case of simle oles 07.31.06.0016.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak a 1 k 1 a j a k b j a k a k 1 1 j,k,j,k 1 j 11 k 1 a j a k 07.31.06.0017.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 j,k,j,k 1 j 11 k 1 a j a k 07.31.06.0018.01 1F a 1,, a 1 ; b 1,, b ; a k a k b j 1 1 ak a j 1 a 1 k 1 a j a k b k 1 1 ak i 0 1 j,k,j,k 1 j 11 k 1 a j a k 07.31.06.0019.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak b j a k a k a k 1 a k b j 1 a k b j a k 1 ak a j 1 a k a j i 0 Res 0; 1 a 1,, 1 a 1 ; ; 1 b 1,, 1 b ; a 1 k 1 a j a k b j a k a k a k i a k b j 1 i 1 i ak a j 1 i i 1 a k, 1, i; ; ; ; 1F a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 ; 0, 1 j,k,j,k 1 j 11 k 1 a j a k Case of oles of order r i the oits a r k ; r, 3, 4 k 07.31.06.000.01 1F a 1,, a 1 ; b 1,, b ; b k ower a 1,, a ; F b 1,, b ;,, ; 0, 1 a k a k1 k r a k a 1 r 1 k 1 j,k,j,k r1 j 1r1 k 1 a j a k r, 3, 4 The major terms for exasios of fuctio 1 F a 1,, a 1 ; b 1,, b ; at
htt://fuctios.wolfram.com 1 07.31.06.001.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak j,k,j,k 1 j 11 k 1 a j a k a 1 k 1 a j a k b j a k a k 1 O 1 ; 07.31.06.00.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 k r1 a 1 k a j a k b j a k a k 1 O 1 r1 Res 0; a 1,, a 1 ; a ; b 1,, b ; j1, j 1, 0; 1 O 1 j ; a k a k1 k r a k a 1 r 1 k 1 j,k,j,k r1 j 1r1 k 1 a j a k r, 3, 4 07.31.06.003.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 a 1 k ak a 1 b k a 1 a 1 1 O 1 1 a a 1 1 a k 3 ak a a a 1 b k a a log Ψa a 1 1 1 Ψa Ψa k a Ψb k a k 3 1 O 1 a 1 k 1 a j a k k 3 b j a k a k 1 O 1 ; a a 1 a k a 1 3 k 1 j,k,j,k 3 j 13 k 1 a j a k
htt://fuctios.wolfram.com 13 07.31.06.004.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 a 1 k ak a 1 b k a 1 a 1 1 O 1 1 a a 1 1 a k 3 ak a a a 1 b k a a log Ψa a 1 1 1 Ψa Ψa k a Ψb k a k 3 1 O 1 1 a a 1 1 a 3 k 4 ak a 3 a 3 a a 3 a 1 b k a 3 a 3 1 log Ψa 3 a 1 1 Ψa 3 a 1 Ψa 3 Ψa k a 3 Ψb k a 3 k 4 5 Π 6 Ψ1 a 3 a 1 1 Ψ 1 a 3 a 1 Ψ 1 a 3 Ψ 1 a k a 3 Ψ 1 b k a 3 1 k 4 O 1 1 a 1 k 1 a j a k a k k 4 b j a k 1 O 1 ; a a 1 a 3 a a k a 1 4 k 1 j,k,j,k 4 j 14 k 1 a j a k
htt://fuctios.wolfram.com 14 07.31.06.005.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 a 1 k ak a 1 b k a 1 a 1 1 O 1 1 a a 1 1 a k 3 ak a a a 1 b k a a log Ψa a 1 1 1 Ψa Ψa k a Ψb k a k 3 1 O 1 1 a a 1 1 a 3 k 4 ak a 3 a 3 a a 3 a 1 b k a 3 a 3 1 log Ψa 3 a 1 1 Ψa 3 a 1 Ψa 3 Ψa k a 3 Ψb k a 3 k 4 5 Π 6 Ψ1 a 3 a 1 1 Ψ 1 a 3 a 1 Ψ 1 a 3 Ψ 1 a k a 3 Ψ 1 b k a 3 1 k 4 O 1 1 1 a 1 a a 3 a 1 4 a 4 k 5 ak a 4 6 a 4 a 3 a 4 a a 4 a 1 b k a 4 a 4 3 1 log Ψa 4 a 1 1 Ψa 4 a 1 Ψa 4 a 3 1 Ψa 4 Ψa k a 4 Ψb k a 4 k 5 1 3 Ψ 1 a 4 Ψ 1 a 4 a 1 1 Ψ 1 a 4 a 1 Ψ 1 a 4 a 3 1 Ψ 1 a k a 4 Ψ 1 b k a 4 k 5 7 Π log Ψa 4 a 1 1 Ψa 4 a 1 1 Ψa 4 a 3 1 Ψa 4 Ψa k a 4 Ψb k a 4 k 5 1 Ψ a 4 a 1 1 Ψ a 4 a 1 Ψ a 4 a 3 1 Ψ a 4 Ψ a k a 4 Ψ b k a 4 Ζ3 k 5 1 O 1 a 1 k 1 a j a k k 5 b j a k a k 1 O 1 ; a a 1 a 3 a a 4 a 3 a k a 1 5 k 1 j,k,j,k 5 j 15 k 1 a j a k Exasios at for olyomial cases
htt://fuctios.wolfram.com 15 07.31.06.006.01 F, a, a 3,, a ; b 1,, b ; k a k 1F 1, 1 b 1,, 1 b ; 1 a, 1 a 3,, 1 a ; 11 b k ; Asymtotic series exasios Exasios for 07.31.06.007.01 F a 1,, a ; b 1,, b ; b j ower a 1,, a ; F b 1,, b ;,, ex a 1,, a ; F b 1,, b ;,, ; 07.31.06.008.01 b j F a 1,, a ; b 1,, b ; a j Χ 1 O 1 b j a j a k a j a k b j a k a k 1 O 1 ; j,k,j,k 1 j 1 k a j a k Exasios for 1 07.31.06.009.01 1 F 1 a 1,, a ; b 1,, b 1 ; b j ower a 1,, a ; F b 1,, b 1 ;,, trig a 1,, a ; F,, b 1,, b 1 ; ; 07.31.06.0030.01 F 1 a 1,, a ; b 1,, b 1 ; 1 b j Π a k Χ Π Χ 1 O 1 Π Χ 1 O 1 1 b j a k a k a j a k 1 b j a k a k 1 O 1 ; j,k,j,k 1 j 1 k a j a k 07.31.06.0031.01 F 1 a 1,, a ; b 1,, b 1 ; 1 b j Π a k Χ cosπ Χ 1 O 1 c 1 siπ Χ 1 O 1 1 b j a k a k a j a k 1 b j a k a k 1 O 1 ; Χ 1 A B 1 1 c 1 1 4 3 A B 1 A B 1 3 16 A a k 1 s1 1 s1 B 1 b k a s a j b s b j j,k,j,k 1 j 1 k a j a k s s
htt://fuctios.wolfram.com 16 Exasios for 07.31.06.003.01 F a 1,, a ; b 1,, b ; b j ower a 1,, a ; F b 1,, b ;,, ex a 1,, a ; F b 1,, b ; j,k,j,k 1 j 1 k a j a k,, ; 07.31.06.0033.01 F a 1,, a ; b 1,, b ; 1Β Π Β a k b j Χ exβ 1Β 1 O 1 1Β b j a k a k a j a k b j a k a k 1 O 1 ; Β 1 Χ 1 Β Β 1 a k b k j,k,j,k 1 j 1 k a j a k Exasios for 0 F 07.31.06.0034.01 0F ; b 1, b ; b 1 b 3 3 Π 3 1 1b 3 1 b 1 3 b 1 3 b 1 b 1 3 b 3 b 1 9 b 3 9 16 3 1 4 6 3 b b 3 1 3 9 b 3 b 1 b 1 3 6 b 3 3 b 17 b 4 b 1 9 b 4 1 b 3 3 b 1 b 4 ; 07.31.06.0035.01 0F ; b 1, b ; b 1 b 3 3 Π 3 1 1b 3 1 b 1 O 1 3 ; Exasios for 0 F 3 07.31.06.0036.01 0F 3 ; b 1, b, b 3 ; b 1 b b 3 4 4 4 Π 3 1 4 3 b 1 b b 3 1 b 1 8 b b 3 1 b 1 1 b 1 b 3 8 b 3 8 b b 3 1 7 1 4 3 1 144 b 4 1 64 3 b 3 b 3 1 b 3 1 8 44 b 8 b 3 3 b 44 b 3 4 b 3 1 b 1 16 048 1 b 3 4 b 3 3 b 4 b 3 40 b 3 1 b 1 b 3 3 1 b 3 1 b 3 11 b 1 144 b 4 144 b 3 4 64 b 3 3 8 b 3 176 b 3 64 b 3 3 b 3 1 8 b 44 b 3 4 b 3 1 16 b 1 b 3 3 1 b 3 1 b 3 11 11 ; 07.31.06.0037.01 0F 3 ; b 1, b, b 3 ; b 1 b b 3 4 4 4 Π 3 1 3 b 4 1 b b 3 1 O 1 4 ;
htt://fuctios.wolfram.com 17 Geeral formulas of asymtotic series exasios 07.31.06.0038.01 F a 1,, a ; b 1,, b ; b j F a 1,, a ; b 1,, b ;,, ; 1 07.31.06.0039.01 F a 1,, a ; b 1,, b ; b j Θ,1 ex a 1,, a ; F b 1,, b ;,, ower a 1,, a ; F b 1,, b ;,,,1 trig a 1,, a ; F b 1,, b 1 ;,, ; 1 Mai terms of asymtotic exasios 07.31.06.0040.01 F a 1,, a ; b 1,, b ; b j res s a j s a j s b j s s a k 1 O 1,1 d 1 Χ cosπ Χ 1 O 1 Θ,1 d Χ Β 1Β 1 O 1 1Β ; Β 1 Χ 1 Β Β 1 a k b k d d 1 1Β Π Β 07.31.06.0041.01 F a 1,, a ; b 1,, b ; c k a k 1 O 1,1 e 1 Χ cosπ Χ 1 O Β 1 Χ 1 Β Β 1 a k e e 1 1Β Π 07.31.06.004.01 Β a k 1 Θ,1 e Χ Β 1Β 1 O 1 b k c k b k j,k,j,k 1 j 1 k a j a k 1F a 1,, a 1 ; b 1,, b ; c 1 O 1 d 1 Ψ 1 O 1 ; a k 1 Ψ b j a j c 1 F a 1,, a 1 ; b 1,, b ; 1 d 1 1Β ; b j a j a k a j b j a k Ψ b k 1 ak Ψ Residue reresetatios
htt://fuctios.wolfram.com 18 07.31.06.0043.01 F a 1,, a ; b 1,, b ; b k a k res s j 0 s a k s b k s s j ; 1 1 1 07.31.06.0044.01 F a 1,, a 1 ; b 1,, b ; b k 1 ak 1 res s j 0 1 s ak s b k s s a k j ; 1 Cotiued fractio reresetatios 07.31.10.0001.01 F a 1,, a ; b 1,, b ; b k 1 a k 1 1 a j 1 b j 1 1 a j 1 b j 1 a j 3 b j a j 3 b j 07.31.10.000.01 F a 1,, a ; b 1,, b ; 1 a k b k 1 k k a j k 1 k b j, k a j k 1 k b j 1 1 Differetial euatios Ordiary liear differetial euatios ad wroskias For the direct fuctio itself The differetial euatio for the fuctio F a 1,, a ; b 1,, b ; has the order max, 1. It has two ( 0, for ) or three ( 0, 1,, for 1) sigular oits. If, the the oit 0 is a regular sigular oit, while is a oregular (essetial) sigular oit; if 1, the all three sigular oits are regular. Reresetatio of fudametal system solutios ear oit 0 for 1 i the geeral case
htt://fuctios.wolfram.com 19 07.31.13.0004.01 1 1 w 1 1 b k w w 1 l 1 a l w 1 d d d d b k 1 w l 1 d d a 1 l w w 1 1 b k w w 1 1 a k w 1 w l 1 a l w a l 0 ; l 1 w c 1 F a 1,, a ; b 1,, b ; c, G,1 1 a 1,, 1 a 0, 1 b k, 1 b 1,, 1 b k1, 1 b k1,, 1 b c, G,1 1 1 a 1,, 1 a 0, 1 b 1,, 1 b k1, 1 b k1,, 1 b, 1 b k 1, c 1 G,1 1 1 1 a 1,, 1 a 0, 1 b 1,, 1 b 07.31.13.0005.01 1 1 w 1 1 b k w w 1 l 1 a l w 1 d d d d b k 1 w l 1 d d a 1 l w w 1 1 b k w w 1 1 a k w 1 w l 1 a l w a l 0 ; w c 1 F a 1,, a ; b 1,, b ; l 1 c k1 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1; ; j,k,j,k 1 j 1 k b j b k b k 07.31.13.0006.01 W F a 1,, a ; b 1,, b ;, 1b 1 F a 1 b 1 1,, a b 1 1; b 1, 1 b 1 b,, 1 b 1 b ;,, 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1;,, 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; 1 Π k1 siπ b k siπ b j b k 1 1 b k,1 1 l 1 1 al b k, Θ 1
htt://fuctios.wolfram.com 0 d d 07.31.13.0001.01 d d b d k 1 l 1 d a l w 0 ; w c 1 F a 1,, a ; b 1,, b ; c k1 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1; ; j,k,j,k 1 j 1 k b j b k b k 07.31.13.0007.01 W F a 1,, a ; b 1,, b ;, 1b 1 F a 1 b 1 1,, a b 1 1; b 1, b 1 b 1,, b 1 b 1;,, 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1;,, 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; cost 1 1 b k,1 1 l 1 1 al b k, Θ 1 ; t lim Ε0 Ε 1 b k W Ε F a 1,, a ; b 1,, b ; Ε, Ε 1b 1 F a 1 b 1 1,, a b 1 1; b 1, b 1 b 1,, b 1 b 1; Ε,, Ε 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1; Ε,, Ε 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; Ε 07.31.13.0008.01 W F a 1,, a ; b 1,, b ;, 1b 1 F a 1 b 1 1,, a b 1 1; b 1, 1 b 1 b,, 1 b 1 b ;,, 1b k F a 1 b k 1,, a b k 1; b k, b 1 b k 1,, b k1 b k 1, b k1 b k 1,, b b k 1;,, 1b F a 1 b 1,, a b 1; b, b 1 b 1,, b 1 b 1; 1 k1 b k 1 b j b k 1 1 b k,1 1 l 1 1 al b k, Θ 1 Reresetatio of fudametal system solutios ear oit 1 for 1 i the geeral case 1,0 Below reresetatio icludes fuctios of two kids. The fuctio G 1,1 1 a 1,, 1 a 1 0, 1 b 1,, 1 b is the iecewise aalytical fuctio with a discotiuity o the uite circle 1. It has sigularity ear oit 1of the form cost 1 Ψ,3 1 O 1, whe 1. The fuctios G 3,3 aalytical fuctios ad are bouded ear oit 1. 1,0 The fuctio G 1,1 1 a 1,, 1 a 1 0, 1 b 1,, 1 b iside of 1ca be rereeted through hyergeometric fuctios defied for all comlex. 0, b k, 1 a 1,, 1 a 1 0, b k, 0, 1 b 1,, 1 b are the
htt://fuctios.wolfram.com 1 d d 07.31.13.000.01 1 d d b d k 1 l 1 d a l w 0 ; 1,0 w c 1 G 1,1 1 a 1,, 1 a 1,3 c 0, 1 b 1,, 1 b k1 G 3,3 0, b k, 1 a 1,, 1 a 1 0, b k, 0, 1 b 1,, 1 b ; 1 1 Ψ b j a j Ψ j,k,j,k 1 j 1 k b j b k b k Reresetatio of fudametal system solutios ear oit for 1 i the geeral case d d 07.31.13.0009.01 d d b d k 1 l 1 d a l w 0 ; w c k a k 1 F 1 a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a a k ; 11 j,k,j,k 1 j 1 k a j a k d d 07.31.13.0003.01 d d b d k 1 l 1 d a l w 0 ; w c k a k 1 F 1 a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a a k ; 11 ; j,k,j,k 1 j 1 k a j a k Trasformatios Products, sums, ad owers of the direct fuctio Products of the direct fuctio 07.31.16.0001.01 F a 1,, a ; b 1,, b ; c r F s Α 1,, Α r ; Β 1,, Β s ; d c k k ; c k d k r Α j k s1 F r k, 1 Β 1 k,, 1 Β s k, a 1,, a ; 1 Α 1 k,, 1 Α r k, b 1,, b ; 1rs1 c s k Β j d k c k c k a j k r1 F s k, 1 b 1 k,, 1 b k, Α 1,, Α r ; 1 a 1 k,, 1 a r k, Β 1,, Β s ; 11 d k b j c k
htt://fuctios.wolfram.com 07.31.16.000.01 F a 1,, a ; b 1,, b ; c r F s Α 1,, Α r ; Β 1,, Β s ; d k m 0 r a j m Α j km c m d km k s b j m Β j km mk m 07.31.16.0003.01 0:;r : a 1,, a ; Α 1,, Α r ; F a 1,, a ; b 1,, b ; c r F s Α 1,, Α r ; Β 1,, Β s ; d F 0:;s c, d : b 1,, b ; Β 1,, Β s ; Idetities Recurrece idetities Distat eighbors with resect to 07.31.17.0001.01 1F a 1,, a 1 ; b 1,, b ; b j k a1,, a 1, b 1,, b F 1 1a 1, a ; a 1 a Ψ k; ; Ψ b j 1 j 3 a j a j Fuctioal idetities Relatios betwee cotiguous fuctios 07.31.17.000.01 b F a, b 1, a 3,, a ; b 1,, b ; a F a 1, b, a 3,, a ; b 1,, b ; a b F a, b, a 3,, a ; b 1,, b ; 0 07.31.17.0003.01 c F a, a,, a ; c, b,, b ; a F a 1, a,, a ; c 1, b,, b ; a c F a, a,, a ; c 1, b,, b ; 0 07.31.17.0004.01 d F a 1,, a ; c 1, d, b 3,, b ; c F a 1,, a ; c, d 1, b 3,, b ; c d F a 1,, a ; c 1, d 1, b 3,, b ; 0 07.31.17.0005.01 ca b F a, b, a 3,, a ; c, b,, b ; ac b F a 1, b, a 3,, a ; c 1, b,, b ; bc a F a, b 1, a 3,, a ; c 1, b,, b ; 0 07.31.17.0006.01 cd a F a, a,, a ; c, d 1, b 3,, b ; dc a F a, a,, a ; c 1, d, b 3,, b ; ac d F a 1, a,, a ; c 1, d 1, b 3,, b ; 0 b k 07.31.17.0007.01 F a, a,, a ; b 1,, b ; F a 1, a,, a ; b 1,, b ; a j F a 1, a 1,, a 1; b 1 1,, b 1; 0 j
htt://fuctios.wolfram.com 3 07.31.17.0008.01 cc 1 b k F a 1,, a ; c, b,, b ; F a 1,, a ; c 1, b,, b ; k a j F a 1 1,, a 1; c, b 1,, b 1; 0 07.31.17.0009.01 b k F a, b 1, a 3,, a ; b 1,, b ; F a 1, b, a 3,, a ; b 1,, b ; b a a j F a 1, b 1, a 3 1,, a 1; b 1 1,, b 1; 0 j 3 07.31.17.0010.01 cc 1 b k F a, a,, a ; c, b,, b ; F a 1, a,, a ; c 1, b,, b ; k c a a j F a 1, a 1,, a 1; c, b 1,, b 1; 0 j 07.31.17.0011.01 b k c F a, b, a 3,, a ; c, b,, b ; k a F a 1, b 1, a 3,, a ; c 1, b,, b ; c a F a, b 1, a 3,, a ; c 1, b,, b ; a a j F a 1, b 1, a 3 1,, a 1; c 1, b 1,, b 1; 0 j 3 07.31.17.001.01 F a 1, b 1, a 3,, a ; c 1, d 1, e 1, b 4,, b ; c da eb e a b c ed e F a, b, a 3,, a ; c, d, e 1, b 4,, b ; c ea db d a b c de d F a, b, a 3,, a ; c, d 1, e, b 4,, b ; d ea cb c a b d ce c F a, b, a 3,, a ; c 1, d, e, b 4,, b ; 0 07.31.17.0013.01 a bd ce c F a, b, c, a 4,, a ; d, e, b 3,, b ; d e a cb c F a 1, b 1, c, a 4,, a ; d 1, e 1, b 3,, b ; a cd be b d e a bc b F a 1, b, c 1, a 4,, a ; d 1, e 1, b 3,, b ; b cd ae a d e b ac a F a, b 1, c 1, a 4,, a ; d 1, e 1, b 3,, b ; 0
htt://fuctios.wolfram.com 4 07.31.17.0014.01 a a j1 b j 1 F a, a,, a 1 ; b 1,, b ; b j a b j b j b k k j b j a k1 1 F a, a,, a 1 ; b 1,, b j1, b j 1, b j1,, b ; a1 1 F a 1, a,, a 1 ; b 1,, b ; Relatios of secial kid 07.31.17.0015.01 F a 1,, a ; c, 1 c, b 3,, b ; F a 1,, a ; c, 1 c, b 3,, b ; F a 1,, a ; 1 c, 1 c, b 3,, b ; 07.31.17.0016.01 F a, a,, a ; a, 1 a, b 3,, b ; F a, a,, a ; 1 a, 1 a, b 3,, b ; 1 F 1 a,, a ; 1 a, b 3,, b ; 07.31.17.0017.01 F a, a,, a ; 1 a, b,, b ; F a, a,, a ; 1 a, b,, b ; 1 F 1 a, a, a,, a ; 1 a, 1 a, b,, b ; 07.31.17.0018.01 F a, 1 a, a 3,, a ; b 1,, b ; F a, 1 a, a 3,, a ; b 1,, b ; 1 F 1 a, a, a 3,, a ; b 1,, b ; Divisio o eve ad odd arts ad geeraliatio 07.31.17.0019.01 F a 1,, a ; b 1,, b ; A A ; A 1 F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; A 1 F a 1,, a ; b 1,, b ; F a 1,, a ; b 1,, b ; 07.31.17.000.01 F a 1,, a ; b 1,, b ; A A ; a A 1 a F 1,,, a 1 1 a 1,, A a j a 1 1 F 1 b j a 1,, ; 1, b 1,, b, a 1,,, b 1 1 a b 1,, ; 4 1 ; 3, b 1 1,, b 1, b 1,, b ; 4 1 07.31.17.001.01 F a 1,, a ; b 1,, b ; 1 k k a j k 1 F 1, a 1 k b j k,, a 1 k 1,, a k,, a k 1 ; k 1,, k, b 1 k,, b 1 k 1,, b k,, b k 1 ; 1 Case 1 F
htt://fuctios.wolfram.com 5 07.31.17.00.01 1F a 1,, a 1 ; b 1,, b ; b k 1 ak a 1 k 1 a j a k b j a k a k 1F a k, a k b 1 1,, a k b 1; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1 ; m 0, 1 j,k,j,k 1 j 11 k 1 a j a k 07.31.17.003.01 m b j b k 1 a j b k 1 j 1 b k 1 a j b k j m1 1 b j b k 1F 1 a 1 b k,, 1 a 1 b k ; 1 b 1 b k,, 1 b k1 b k, 1 b k1 b k,, 1 b 1 b k ; 1 m1 ak a j m 1 a k b j 1 j m1 a k 1 1 a k b j j 1 a j a k 1 1F 1 a k b 1,, 1 a k b 1 ; 1 a 1 a k,, 1 a k1 a k, 1 a k1 a k,, 1 a 1 a k ; 1m1 m m 1 1 m m 1, 0 ; 07.31.17.004.01 a 1 1F a 1,, a 1 ; b 1,, b ; w 1 a k 1 1 F k, a,, a 1 ; b 1,, b ; w k 1 k Differetiatio Low-order differetiatio With resect to a 1 07.31.0.0001.01 F 1,0,,0,0,,0,0 a 1,, a ; b 1,, b ; 1 1 07.31.0.000.01 F 1,0,,0,0,,0,0 a 1,, a ; b 1,, b ; Ψk a 1 a j k k j a j k b j k F 1 0 1 b j Ψa 1 F a 1,, a ; b 1,, b ; ; 1 a 1 1,, a 1; 1; 1, a 1 ;, b 1 1,, b 1;; a 1 1;, With resect to b 1
htt://fuctios.wolfram.com 6 07.31.0.0003.01 F 0,,0,1,0,,0,0 a 1,, a ; b 1,, b ; Ψb 1 F a 1,, a ; b 1,, b ; 1 1 07.31.0.0004.01 F 0,,0,1,0,,0,0 a 1,, a ; b 1,, b ; a j F 1 0 1 b 1 b j Ψk b 1 a j k k k b j k 1 a 1 1,, a 1; 1; 1, b 1 ;, b 1 1,, b 1;; b 1 1;, With resect to elemet of arameters With resect to elemet of arameters 07.31.0.0005.01 F a, a,, a ; a 1, b,, b ; a j a j a 1 1 F 1 a 1, a 1, a 1,, a 1; a, a, b 1,, b 1; j b j 07.31.0.0006.01 F a 1, a,, a ; a, b,, b ; a With resect to 07.31.0.0007.01 F a 1,, a ; b 1,, b ; 07.31.0.0008.01 F a 1,, a ; b 1,, b ; a j j a j a 1 F 1 a 1,, a 1; b 1,, b 1; j b j F a 1 1,, a 1; b 1 1,, b 1; b j a j a j 1 F a 1,, a ; b 1,, b ; b j b j 1 ; Symbolic differetiatio With resect to a 1 07.31.0.0009.01 F,0,,0,0,,0,0 a 1,, a ; b 1,, b ; j a j k a 1 k k b j a k 1 k ; 1 1 With resect to b 1 07.31.0.0010.01 F 0,,0,,0,,0,0 a 1,, a ; b 1,, b ; a j k k j b j k 1 b 1 k With resect to elemet of arameters With resect to elemet of arameters b 1 k ; 1 1 1
htt://fuctios.wolfram.com 7 07.31.0.001.01 F a, a,, a ; a 1, b,, b ; a 1 1 j a j a 1 1 j b j 07.31.0.00.01 F a 1, a,, a ; a, b,, b ; a F a 1,, a 1, a 1,, a 1; a,, a, b 1,, b 1; ; 1 a 1 F 1 a,, a ; b,, b ; a 1 j a j 1 1 1 F 1 a,, a ; b,, b ; ; a With resect to 07.31.0.0011.01 F a 1,, a ; b 1,, b ; a j 1F 1 a 1,, a 1; b 1,, b 1; j b j F a 1,, a ; b 1,, b ; ; b j 07.31.0.001.01 F a 1,, a ; b 1,, b ; b j 1 F 11, a 1,, a ; 1, b 1,, b ; ; 07.31.0.0013.01 Α F a 1,, a ; b 1,, b ; 1 Α Α 1F 1 Α 1, a 1,, a ; Α 1, b 1,, b ; ; 07.31.0.0014.01 a1 F a, a,, a ; b 1,, b ; a a1 F a, a,, a ; b 1,, b ; ; 07.31.0.0015.01 c1 F a 1,, a ; c, b,, b ; c c1 F a 1,, a ; c, b,, b ; ; 07.31.0.0016.01 F, a,, a ; 1, b,, b ; 1 F 1, 1, a,, a ; 1, 1, b,, b ; ; 07.31.0.0017.01 Α F, a,, a ; b 1,, b ; 1 Α Α 1 F 1, Α 1, a,, a ; Α 1, b 1,, b ; ;
htt://fuctios.wolfram.com 8 07.31.0.0018.01 Α F r, 1 r,, r1, a r r1,, a ; b 1,, b ; m 1 Α Α m F m 1 r 1,,,, Α 1 r r r m, Α Α m,, m m, a r1, Α 1 Α Α m, a ;,,,, b 1,, b ; m ; r m m m m 07.31.0.0019.01 F, a,, a ; b 1,, b ; 1 k k 1 F, k, a k,, a k; b 1 k,, b k; ; k b j k Fractioal itegro-differetiatio With resect to 07.31.0.000.01 Α F a 1,, a ; b 1,, b ; Α b Α j 1 F 11, a 1,, a ; 1 Α, b 1,, b ; Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio 07.31.1.0001.01 F a 1,, a ; b 1,, b ; b j 1 F a 1 1,, a 1; b 1 1,, b 1; a j 1 Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig ower fuctio 07.31.1.000.01 Α1 F a 1,, a ; b 1,, b ; Α Α 1F 1 Α, a 1,, a ; Α 1, b 1,, b ; Defiite itegratio For the direct fuctio itself
htt://fuctios.wolfram.com 9 07.31.1.0003.01 0 t Α1 F a 1,, a ; b 1,, b ; tt b k Α a k Α 0 ReΑ mirea 1,, Rea 1 0 ReΑ mi Rea 1,, Rea, 1 4 1 Re a k b k Α ; a j b k 1 Summatio Fiite summatio 1 07.31.3.0001.01 a j k 1 F 1, a 1 k b j k,, a 1 k 1,, a k,, a k 1 ; k 1 k,,, b 1 k,, b 1 k 1, b k,, b k 1 ; 1 k k F a 1,, a ; b 1,, b ; Ifiite summatio 07.31.3.000.01 a 1 k 1 k 1 F k, a,, a 1 ; b 1,, b ; w k 1 a 1 w 1F a 1,, a 1; b 1,, b ; 1 Oeratios Limit oeratio 07.31.5.0001.01 lim 1 Ψ 1 F a 1,, a 1 ; b 1,, b ; 1 lim 1 07.31.5.000.01 1 log1 1 F a 1,, a 1 ; b 1,, b ; 07.31.5.0003.01 F a 1,, a ; b 1,, b ; lim b 1 b 1 Ψ b j 1 a j b j 1 a j ; Ψ b j a j ReΨ 0 1 ; Ψ b j a j Ψ 0 1 a j 1 F a 1 1,, a 1;, b 1,, b 1; ; 07.31.5.0004.01 1 lim a F a, a,, a ; b 1,, b ; a 1F a,, a ; b 1,, b ;
htt://fuctios.wolfram.com 30 07.31.5.0005.01 lim a F a 1,, a ; b, b,, b ; b F 1 a 1,, a ; b,, b ; lim a 07.31.5.0006.01 a 1 a 1 F a, a, a 3,, a 1 ; a 1, a 3 1,, a 1 1; 1 1 1 S a a 3 a 1 a 07.31.5.0007.01 lim 1 1 F 1 m, a 1, a 3 1,, a 1 1; a, a 3,, a 1 ; 1 m1 m m 1 ; a a 3 a 1 1 m ; Reresetatios through more geeral fuctios Through hyergeometric fuctios Ivolvig F 07.31.6.0001.01 F a 1,, a ; b 1,, b ; b k F a 1,, a ; b 1,, b ; Through hyergeometric fuctios of two variables 07.31.6.000.01 0 0 F a 1,, a ; b 1,, b ; F ; a 1,, a ;; 0 0 ; b 1,, b ;;, 0 07.31.6.0003.01 F a 1,, a ; b 1,, b ; b k F 0 0 ; a 1,, a ;; 0 0 ; b 1,, b ;;, 0 Through Meijer G Classical cases for the direct fuctio itself 07.31.6.0004.01 F a 1,, a ; b 1,, b ; b k a k 1, G,1 1 a 1,, 1 a 0, 1 b 1,, 1 b
htt://fuctios.wolfram.com 31 07.31.6.0005.01 1F a 1,, a 1 ; b 1,, b ; b k 1 Π siψ Π ak 1 siπ b j a k siπ b j b k k j,1 G 1,1 1 a 1,, 1 a 1 0, 1 b j, 1 b 1,, 1 b j1, 1 b j1,, 1 b Π b k 1 siψ Π ak 1 Ψ 1 Ψ 0,1 G 1,1 1 a 1,, 1 a 1 1,0 0, 1 b 1,, 1 b G 1,1 1 a 1,, 1 a 1 0, 1 b 1,, 1 b ; Ψ b j a j 1, 0 Ψ 1 07.31.6.0006.01 F a 1,, a ; b 1,, b ; b k a k G 3,1 4,3 1 1, b 1,, b a 1,, a ; 0, Theorems Coectios betwee series ad cotiued fractio reresetatios Euler established that the coverget series a k ca be exressed i a cotiued fractio form a k a 1 1 CotiueFractio a k1 a k, 1 a 1 k1, k, 1,. a k I articular the followig reresetatio takes lace: F a 1, a,, a ; b 1, b,, b ; 1 a k b k 1 CotiueFractio k a j k 1 k b j, 1 k a j k 1 k b j, k, 1, 1. History J. F. Pfaff (1797) T. Clause (188); J. Thomae (1870, 1879) studied differetial euatio S. Picherle (1886, 1888) L. Pochhammer (1888) E. W. Bares (1906 1908) T. W.Chaudy (1943) N. E. Nörlud (1955) A. P. Prudikov, Y.A. Brychkov ad O.I. Marichev (1986)
htt://fuctios.wolfram.com 3 Coyright This documet was dowloaded from fuctios.wolfram.com, a comrehesive olie comedium of formulas ivolvig the secial fuctios of mathematics. For a key to the otatios used here, see htt://fuctios.wolfram.com/notatios/. Please cite this documet by referrig to the fuctios.wolfram.com age from which it was dowloaded, for examle: htt://fuctios.wolfram.com/costats/e/ To refer to a articular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: htt://fuctios.wolfram.com/01.03.03.0001.01 This documet is curretly i a relimiary form. If you have commets or suggestios, lease email commets@fuctios.wolfram.com. 001-008, Wolfram Research, Ic.