Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 7, Numbe 1, pp 41 55 (2012 http://campusmstedu/adsa -analogues of Tiple Seies Reduction Fomulas due to Sivastava and Panda with Geneal Tems Thomas Enst Uppsala Univesity Depatment of Mathematics PO Box 480, SE-751 06 Uppsala, Sweden thomas@mathuuse Abstact We find six double -analogues of the eduction fomulas by Sivastava and Panda fo tiple seies, with geneal tems These ae then spezialized to summations fo -Kampé de Féiet functions AMS Subject Classifications: 33D15, 33D05, 33C65 Keywods: Tiple seies, geneal tems, eduction fomula, Γ -function 1 Intoduction The histoy of the theme in this aticle goes many yeas back It deals with eualities between uite geneal tiple and double sums The coefficients consist of -shifted factoials, Γ -functions and -binomial coefficients Sivastava and Panda [10] collected these fomulas, peviously scatteed in the liteatue into 6 fomulas of a slightly diffeent chaacte than we pesent hee (fo 1 In Section 2 we find double -foms of these, by the 2 -Vandemonde summation fomulas, ie, 12 -foms These ae then specialized, in Section 3, to six summation fomulas fo -Kampé de Féiet functions We will use the (; l; λ opeato, which is exploed in detail in [6] We tun to the fist definitions Definition 11 The powe function is defined by a e a log( Let δ > 0 be an abitay small numbe We will use the following banch of the logaithm: π + δ < Im (log π + δ This defines a simply connected space in the complex plane The vaiables a, b, c, C denote cetain paametes The vaiables i, j, k, l, m, n, p, will denote natual numbes except fo cetain cases whee it will be clea fom Received July 1, 2010; Accepted Novembe 22, 2011 Communicated by Lance Littlejohn
42 Thomas Enst the context that i will denote the imaginay unit Let the -shifted factoial be defined by 1, n 0; n 1 a; n (1 a+m (11 n 1, 2, m0 Since poducts of -shifted factoials occu so often, to simplify them we shall feuently use the moe compact notation The opeato is defined by By (13 it follows that a 1,, a m ; n : C Z C Z a a + m a j ; n (12 j1 πi log (13 n 1 a; n (1 + a+m, (14 m0 Assume that (m, l 1, ie, m and l elatively pime The opeato m l : C Z C Z is defined by a a + 2πim l log (15 We will also need anothe genealization of the tilde opeato n 1 k 1 k ã; n ( i(a+m (16 m0 i0 This leads to the following -analogue of [11, p22, (2] Theoem 12 (See [4] a; kn k 1 a + m a ; n k + m ; n (17 k k m0 The following notation will be convenient: QE(x x
-analogues of Tiple Seies Reduction Fomulas 43 Definition 13 (; l; λ; n l 1 λ + m λ ; n l + m ; n (18 l l m0 When λ is a vecto, we mean the coesponding poduct of vecto elements When λ is eplaced by a seuence of numbes sepaated by commas, we mean the coesponding poduct as in the case of -shifted factoials Definition 14 Let have dimensions Let (a, (b, (g i, (h i, (a, (b, (g i, (h i A, B, G i, H i, A, B, G i, H i 1 + B + B + H i + H i A A G i G i 0, i 1,, n Then the genealized -Kampé de Féiet function is defined by Φ A+A :G 1 +G 1 ;;Gn+G n B+B :H 1 +H 1 ;;Hn+H n m [ ] (a ˆ : (gˆ 1 ; ; (gˆ n (b ˆ : (hˆ 1 ; ; (hˆ n ; x (a : (g 1; ; (g n (b : (h 1; ; (h n (a; ˆ 0 m (a ( 0, m n j1 ( (g ˆ j ; j mj ((g j( j, m j x m j j (b; ˆ 0 m (b ( 0, m n j1 ( (h ˆ j ; j mj (h j ( j, m j 1; j mj ( 1 n j1 m j(1+h j +H j G j G j +B+B A A ( ( m n ( QE (B + B A A, 0 QE (1 + H j + H j G j G 2 j We assume that no factos in the denominato ae zeo We assume that j1 (a ( 0, m, (g j( j, m j, (b ( 0, m, (h j( j, m j contain factos of the fom ˆ a(k; k, (s; k, (s(k; k o QE (f( m ( mj 2, j (19 Definition 15 Let the Gauss -binomial coefficient be defined by ( n 1; n, k 0, 1,, n (110 k 1; k 1; n k Let the Γ -function be defined in the unit disk 0 < < 1 by Γ (x 1; x; (1 1 x (111
44 Thomas Enst 2 Cetain -summation Fomulas Sivastava [12] and Panda & Sivastava [10] have systematically collected and genealized a numbe of elated summation fomulas known fom the liteatue Ou task in this section is to find symmetic -analogues of these fomulas, which always occu in pais In cetain exceptional cases the convegence in the fomulas is not so good, we then eplace the euality sign by the sign fo fomal euality, We assume thoughout that M km + ln and {} is a seuence of bounded complex numbes Theoem 21 (Compae [10, (4 p 244] and [12, (9 p 28] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N 1 β+; M ( 2 M xm y n 1 β; M NM 1 + α β; M 1 + α β N; M (21 Poof We have LHS Γ (α Γ (α ( 1 Γ (α Γ (β 1 β+; M ( 2 M 1 β; M ( α+β M+N Γ (α β α; N Γ (α 1 α; N 1 β; +M 1 α; N; 1; ( α+β M+N 1 β + M; 1 α; N; 1; xm y n 1 β; M α + β M; N 1 α; N xm y n 1 β; M NM 1 + α β; M 1 + α β N; M (22 The poof is complete
-analogues of Tiple Seies Reduction Fomulas 45 Theoem 22 (Compae [10, (4 p 244] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N 1 β+; M ( 2+(α β N+1 xm y n 1 β; M N(1 β 1 + α β; M 1 + α β N; M (23 Poof We have LHS ( 1 Γ (α Γ (β (( QE + (α β N + 1 2 Γ (α Γ (α β α; N Γ (α 1 α; N Γ (α 1 β; M 1 β+; M 1 β; +M 1 α; N; 1; 1 β + M; 1 α; N; 1; xm y n 1 β; M α + β M; N 1 α; N N(1 β+m xm y n 1 β; M N(1 β 1 + α β; M 1 + α β N; M (24 The poof is complete Theoem 23 (Compae [10, (5 p 244] and [12, (8 p 28] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N 1 β+; M ( 2 1 α+; M xm y n 1 β; M 1 1 α + N; M (25
46 Thomas Enst Poof We have LHS Γ (α Γ (α ( 1 Γ (α Γ (β ( α+β+n Γ (α β α; N Γ (α 1 α; N 1 β+; M ( 2 1 α+; M 1 β; M 1 α; M 1 β; +M 1 α; +M N; 1; ( α+β+n 1 β + M; 1 α + M; N; 1; xm y n 1 β; M α + β; N 1 α; M 1 α + M; N xm y n 1 β; M 1 1 α + N; M (26 The poof is complete Theoem 24 (Compae [10, (5 p 244] ( 1 Γ (α 1 β+; M Γ (β 1 α+; M (( QE + (α β N + 1 2 β α; N Γ (α xm y n 1 β; M N(1 β+m 1 α; N 1 α + N; M (27
-analogues of Tiple Seies Reduction Fomulas 47 Poof We have LHS ( 1 Γ (α Γ (β (( QE + (α β N + 1 2 Γ (α 1 β; +M N; Γ (α 1 α; +M 1; 1 β + M; N; Γ (α 1 α + M; 1; 1 β+; M 1 α+; M 1 β; M 1 α; M xm y n 1 β; M α + β; N N(1 β+m 1 α; M 1 + α + M; N β α; N Γ (α xm y n 1 β; M N(1 β+m 1 α; N 1 α + N; M (28 The poof is complete Theoem 25 (Compae [10, (6 p 244] and [12, (10 p 29] ( 1 Γ (α α ; M Γ (β β ; M β α; NΓ (α NM α N; M 1 α; N β; M QE (( 2 (29
48 Thomas Enst Poof We have LHS ( 1 Γ (α Γ (β Γ (α α; M Γ (α β; M ( α+β+n α; M Γ (α β; M β α; N Γ (α 1 α; N The poof is complete α ; M ( 2 β ; M α; M β; M N; 1; N 1 β M; N; 1 α M; 1; α + β; N 1; m 1; n 1 α M; N NM α N; M β; M Theoem 26 (Compae [10, (6 p 244] ( 1 Γ (α α ; M ( Γ (β β ; M β α; NΓ (α xm y n N(1 β α N; M 1 α; N β; M Poof We have LHS ( 1 Γ (α Γ (β Γ (α α; M Γ (α β; M α; M Γ (α β; M β α; N Γ (α 1 α; N 2+(α β N+1 α ; M ( β ; M α; M β; M N; 1; (α β+1 (210 (211 2+(α β N+1 1 β M; N; 1 α M; 1; α + β; N N(1 β M 1 α M; N α N; M N(1 β β; M (212
-analogues of Tiple Seies Reduction Fomulas 49 The poof is complete Theoem 27 (Compae [10, (7 p 244] and [12, (11 p 29] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N Poof We have LHS Γ (α Γ (α ( 1 Γ (α Γ (β β; M ( 2 β ; M ( 2 β; M α + β + N; M α + β; M ( α+β+m+n Γ (α β α; N Γ (α 1 α; N ( 2 β ; M N; α ; 1; N β; M 1 β M; 1 α; N; 1; β; M α + β + M; N 1 α; N β; M α + β + N; M α + β; M (213 (214 The poof is complete Remak 28 The convegence in fomula (213 is not so good Howeve, this fomula woks well in a numbe of special cases One example is 1, 85, α 53, β 5543, N 4, x 2, y 176 (215 In this case the diffeence LHS-RHS in (213 is 409273 10 12 fo 0 m 60, 0 n 60 Theoem 29 (Compae [10, (7 p 244] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N (2+(1+α β M N β ; M N(1 β M β; M α + β + N; M α + β; M (216
50 Thomas Enst Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α β; M Γ (α β α; N Γ (α 1 α; N The poof is complete β ; M (1+α β M N ( 2+1 2M N; α ; 1; (1+α β M β; M 1 β M; 1 α; N; 1; N(1 β M β; M α + β + M; N 1 α; N N(1 β M β; M α + β + N; M α + β; M Theoem 210 (Compae [10, (8 p 244] and [12, (12 p 29] ( 1 Γ (α α ; M Γ (β β α; N Γ (α 1 α; N Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α Γ (α QE xm y n α N; M 1 + α β; M 1 + α β N; M α ; M ( 2 α; M β α; N Γ (α 1 α; N (217 (( 2 β ; α; M N; 1; N α; M α + β M; N 1 M α; N (218 1 β; ( α+β M+N 1 α M; N; 1; α N; M 1 + α β; M 1 + α β N; M (219
-analogues of Tiple Seies Reduction Fomulas 51 The poof is complete Theoem 211 (Compae [10, (8 p 244] ( 1 Γ (α ( 2+(1+α β+m N α ; M Γ (β 1; m 1; n β α; N Γ (α xm y n 1 β; M N(1 β 1 + α β; M 1 α; N 1 + α β N; M Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α Γ (α α; M β α; N Γ (α 1 α; N (220 α ; M ( 2+(1+α β+m N β ; α; M N; (1+α β+m 1; 1 β; 1 α M; N; 1; α; M N(1 β α + β M; N 1 M α; N The poof is complete α N; M N(1 β 1 + α β; M 1 + α β N; M (221 Remak 212 The convegence in fomula (220 is not so good Howeve, this fomula woks somehow in a numbe of special cases One example is 1, 99, α 43, β 5543, N 4, x 1, y 076 (222 In this case the diffeence LHS-RHS in (220 is 00000127104 fo 0 m 20, 0 n 20 Theoem 213 (Compae [10, (9 p 244] and [12, (13 p 29] ( 1 Γ (α ( 2+M Γ (β 1 α+; M β α; N Γ (α α + β + N; M 1 α; N 1 α + N, α + β; M (223
52 Thomas Enst Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α ( α+β+m+n Γ (α β α; N Γ (α 1 α; N The poof is complete 1 α; M ( 2+M 1 α+; M 1 β; 1 α; +M N; 1; ( α+β+m+n 1 β; 1 α + M; N; 1; 1 α; M α + β + M; N 1 α + M; N α + β + N; M 1 α + N, α + β; M Theoem 214 (Compae [10, (9 p 244] ( 1 Γ (α ( 2+(α β+1 N Γ (β 1 α+; M β α; N Γ (α NM α + β + N; M N(1 β 1 α; N 1 α + N, α + β; M Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α Γ (α β α; N Γ (α 1 α; N The poof is complete 1 α; M ( 2+(α β+1 N 1 α+; M 1 β; 1 α; +M N; 1; 1 β; 1 α + M; N; 1; 1 α; M N(1 β α + β + M; N 1 α + M; N xm y n N(1 β α + β + N; M 1 α + N, α + β; M (224 (225 (226
-analogues of Tiple Seies Reduction Fomulas 53 3 Specializations Specializing the pevious fomulas leads to the following fomulas fo two vaiables, whee we have put θ(n; α, β; ; ( 1 ( 2 Γ (α Γ (β ; ω(n; α, β; β α; NΓ (α 1 α; N (31 We have assumed that B B G G in all fomulas The conditions fo A and E ae given sepaately in each case Theoem 31 (Compae [10, (16 p 246] We assume that A + 2l E [ ] (; l; 1 β+, (a : (b; (b θ(n; α, β; ; Φ A+2l:B E:G (e : (g; (g ; x l(n, y l(n [ ] (; l; 1 β, 1 + α β, (a : (b; (b ω(n; α, β; Φ A+4l:B (; l; 1 + α β N, (e : (g; (g ; x, y (32 Poof Put (a; m+n (b; m (b ; n (e; m+n (g; m (g ; n Nl(m+n (33 in (21 We have assumed that k l In the following poofs we use the value and assume that k l (a; m+n (b; m (b ; n (e; m+n (g; m (g ; n (34 Theoem 32 (Compae [10, (19 p 247] We assume that A E [ ] (; l; 1 β+, (a : (b; (b θ(n; α, β; ; Φ A+2l:B (; l; 1 α+, (e : (g; (g ; x, y [ ] ω(n; α, β; Φ A+2l:B (; l; 1 β, (a : (b; (b (; l; 1 α + N, (e : (g; (g ; x, y Poof Use (25 Theoem 33 (Compae [10, (20 p 247] We assume that A E [ ] (; l; α, (a : (b; (b θ(n; α, β; ; Φ A+2l:B (; l; β, (e : (g; (g ; x, y [ ] (; l; α N, (a : (b; (b ω(n; α, β; Φ A+2l:B (; l; β, (e : (g; (g ; x Nl, y Nl (35 (36
54 Thomas Enst Poof Use (29 Theoem 34 (Compae [10, (21 p 247] We assume that A E + 2l [ θ(n; α, β; ; (1+α β N Φ A:B ω(n; α, β; N(1 β Φ A+2l:B E+4l:G Poof Use (216 ] (a : (b; (b (; l; β, (e : (g; (g ; x l, y l [ (; l; β α + N, (a : (b; (b (; l; β, α + β, (e : (g; (g ; x Nl, y Nl (37 ] Theoem 35 (Compae [10, (22 p 248] We assume that A + 2l E θ(n; α, β; ; Φ A+2l:B E:G ω(n; α, β; Φ A+4l:B [ (; l; α, (a : (b; (b (e : (g; (g [ (; l; α N, 1 + α β, (a : (b; (b (; l; 1 + α β N, (e : (g; (g ] ; x, y ] ; x, y (38 Poof Use (218 Remak 36 In [9, (31 p 439] Kandu tied to deive a simila fomula Howeve, in this aticle the definitions ae insufficient Remak 37 Fomula (38 is a -analogue of [8] Theoem 38 (Compae [10, (23 p 248] We assume that A E + 2l [ θ(n; α, β; ; Φ A:B ω(n; α, β; Φ A+2l:B E+4l:G [ ] (a : (b; (b (; l; 1 α+, (e : (g; (g ; xl, y l (; l; α + β + N, (a : (b; (b (; l; 1 α + N, α + β, (e : (g; (g ] ; x, y (39 Poof Use (223 4 Conclusion We expect that these fomulas will be of geatest value when looking fo -analogues of eductions fo tiple -seies This is an investigation which has only stated, and hopefully will continue in the next yeas The connection with Γ -functions is uite inteesting, as is manifested in the book [7]
-analogues of Tiple Seies Reduction Fomulas 55 Refeences [1] T Enst, A method fo -calculus J nonlinea Math Physics 10 No4 (2003, 487-525 [2] T Enst, Some esults fo -functions of many vaiables Rendiconti di Padova 112 (2004, 199-235 [3] T Enst, A enaissance fo a -umbal calculus Poceedings of the Intenational Confeence Munich, Gemany 25-30 July 2005 Wold Scientific, (2007 [4] T Enst, Some new fomulas involving Γ functions Rendiconti di Padova 118 (2007, 159 188 [5] T Enst, The diffeent tongues of -calculus, Poceedings of Estonian Academy of Sciences 57 no 2 (2008, 81-99 [6] T Enst, -analogues of geneal eduction fomulas by Buschman and Sivastava, togethe with the impotant (l; λ opeato eminding of MacRobet Demonstatio Mathematica XLIV (2011, 285-296 [7] T Enst, A compehensive teatment of -calculus, Bikhäuse, 2012 [8] KC Gupta and A Sivastava, Cetain esults involving Kampé de Féiet s function Indian J Math 15 (1973, 99 102 [9] D Kandu, Cetain expansions involving basic hypegeometic functions of two vaiables Indian J Pue Appl Math 18, no 5, (1987, 438 441 [10] R Panda and HMSivastava, Some ecusion fomulas associated with multiple hypegeometic functions Bull Acad Polon Sci Se Sci Math Astonom Phys 1975 23, no 3 (1975, 243 250 [11] ED Rainville, Special functions Repint of 1960 fist edition (Chelsea Publishing Co, Bonx, NY, 1971 [12] H MSivastava, Cetain summation fomulas involving genealized hypegeometic functions Comment Math Univ St Paul 21, (1972/73 25 34 58, 241 245