Some circular summation formulas for theta functions

Transcription

1 Ci et l. Boundr Vlue Prolems 013, 013:59 R E S E A R C H Open Access Some circulr summtion formuls for thet functions Yi Ci, Si Chen nd Qiu-Ming Luo * * Correspondence: luomth007@163.com Deprtment of Mthemtics, Chongqing Norml Universit, Chongqing Higher Eduction Meg Center, Huxi Cmpus, Chongqing, 01331, People s Repulic of Chin Astrct In this pper, we otin some circulr summtion formuls of thet functions using the theor of elliptic functions nd show some interesting identities of thet functions nd pplictions. MSC: Primr 11F7; secondr 33E05; 11F0 Kewords: circulr summtion; elliptic functions; thet functions; thet function identities 1 Introduction Throughout this pper we tke q = e iπ,wherei > 0. The clssicl Jcoi thet functions θ i z, i =1,,3,,redefinedsfollows: θ 1 z = iq 1/ θ z =q 1/ 1 m q mm+1 e m+1iz, 1.1 q mm+1 e m+1iz, 1. θ 3 z = q m e miz, 1.3 θ z = 1 m q m e miz. 1. Recentl, Chn, Liu nd Ng [1] proved Rmnujn s circulr summtion formuls nd derived identities similr to Rmnujn s summtion formul nd connected these identities to Jcoi s elliptic functions. Susequentl, Zeng [] gve generlized circulr summtion of the thet function θ 3 z s follows: + + πs = C 33, ; + πs, θ z, Ci et l.; licensee Springer. This is n Open Access rticle distriuted under the terms of the Cretive Commons Attriution License which permits unrestricted use, distriution, nd reproduction in n medium, provided the originl work is properl cited.

2 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge of 15 where C 33, ;, = m 1 + +m +n 1 + +n =0 m 1,...,m,n 1,...,n = q m 1 + +m +n 1 + +n e km 1 + +m i. A specil cse of formul 1.5 ields the following result see [1, Theorem 3.1]: k + πs = C 33 k,0;0, θ z, 1.6 where C 33 k,0;0,= m 1 + +m k =0 m 1,...,m k = q m 1 + +m k. 1.7 Upon,, n nd k re n positive integer with k = +. More recentl, Liu further otined the generl formuls for thet functions see [3], ut from one min result, Theorem 1 of Liu, we do not deduce our results in the present pper. Mn people reserch the circulr summtion formuls of thet functions nd find more interesting formuls see, for detils, [ 15]. In the present pper, we otin nlogues nd uniform formuls for thet functions θ 1 z, θ z, θ 3 z ndθ z. We now stte our result s follows. Theorem 1 For n positive integer k, n, ndwithk= +, α =1,,β =3,. For, even, we hve θα + + πs = C αβ, ; θβ + πs, θ z. 1.8 For even, n nd odd, we hve + + πs z θ + πs = C 1, ;, θ z, πs z θ + πs θ = C, ;, θ z, 1.10

3 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 3 of 15 where C αβ, ;, =i α q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 αm 1+ +m +β+1n 1 + +n q m 1 + +m +n 1 + +n +m 1+ +m e km 1+ +m + i Proof of Theorem 1 From Jcoi s thet functions , we hve the following properties respectivel: θ 1 z + π = θ 1 z, θ 1 z + π = q 1 e iz θ 1 z,.1 θ z + π = θ z, θ z + π =q 1 e iz θ z,. θ 3 z + π =θ 3 z, θ 3 z + π =q 1 e iz θ 3 z,.3 θ z + π =θ z, θ z + π = q 1 e iz θ z.. From.1-., using the induction, we esil otin θ 1 z + nπ = 1 n q n e niz θ 1 z,.5 θ z + nπ =q n e niz θ z,.6 θ 3 z + nπ =q n e niz θ 3 z,.7 θ z + nπ = 1 n q n e niz θ z..8 Let f z= + + πs z θ β + πs θ α Cse 1. When α =1,β =3. The function f z ecomesofthefollowingform: f z= + + πs z θ + πs θ 1 From.10weesilotin f z= + + πs z θ + πs s=1 + + z θ + + πs z θ + πs s= z θ f z + π= ,.11..1

4 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge of 15 Compring.11nd.1, when is even, we get f z + π=f z..13 B.5nd.7, nd noting tht + = k,weotin f z + π= + + πs + nπ z θ + πs + nπ θ 1 = 1 n q 1 e iz θ 1 Oviousl, when is even, we hve + + πs z θ + πs..1 f z + π=q 1 e iz f z..15 We construct the function f z θ 3.B.13nd.15, we find tht the function f z n elliptic function with doule periods π nd π nd onl hs simple pole t z = π + π in the period prllelogrm. Hence the function denoted C 13, ;,, i.e., we hve f z θ 3 z = C 13, ;,, f z θ 3 z z θ 3 z is is constnt, s, this constnt is f z=c 13, ;, θ 3 z..16 B 1.3,.10 nd.16, we hve + + πs = C 13, ;, + πs q m e miz..17 B 1.1nd1.3, we otin i q m 1,...,m,n 1,...,n = e izm 1 + +m+n 1 + +n + 1 m m m + +m +n 1 + +n +m 1 + +m q e i m 1 + +m+ n 1 + +n e iπsm 1 + +m+n 1 + +n + = C 13, ;, q m e miz..18

5 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 5 of 15 B equting the constnt term of oth sides of.18, we otin C 13, ;, = i q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 m m m + +m +n 1 + +n +m 1 + +m q e i [km 1+ +m + ]..19 Clerl, C 13, ;, =C 13, ;,,.0 where C 13, ;, = i q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 m 1+ +m q m 1 + +m +n 1 + +n +m 1+ +m e km 1+ +m + i..1 In the sme mnner s in Cse 1, we cn otin Cse elow. Cse. When α =,β =3. The function f z ecomesofthefollowingform: f z= + + πs z θ + πs θ From.weesilotin f z= θ + + πs z θ + πs s=1 + θ + z θ θ + + πs z θ + πs s=1 + 1 θ + z θ f z + π= Compring.3nd., when is even, we get..,.3.. f z + π=f z..5

6 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 6 of 15 B.6nd.7, nd noting tht + = k,weotin f z + π= Oviousl, we hve + + πs + nπ z θ + πs + nπ θ = q 1 e iz θ + + πs z θ + πs..6 f z + π=q 1 e iz f z..7 We construct the function f z θ 3.B.5nd.7, we find tht the function f z z θ 3 z is n elliptic function with doule periods π nd π nd onl hs simple pole t z = π + π in the period prllelogrm. Hence the function f z θ 3 is constnt, s, this constnt is z denoted C 3, ;,, i.e., we hve f z θ 3 z = C 3, ;,, f z=c 3, ;, θ 3 z..8 B 1.3,.nd.8, we hve θ + + πs = C 3, ;, B 1.nd1.3, we otin q m 1,...,m,n 1,...,n = e i m 1 + +m+ = C 3, ;, + πs q m e miz..9 m 1 + +m +n 1 + +n +m 1 + +m q e izm 1 + +m+n 1 + +n + n 1 + +n e iπsm 1 + +m+n 1 + +n + q m e miz..30 B equting the constnt term of oth sides of.30, we otin C 3, ;, =q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 m 1 + +m +n 1 + +n +m 1 + +m q e i [km 1+ +m + ]..31

7 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 7 of 15 Clerl, where C 3, ;, =C 3, ; C 3, ;, =q e i,,.3 q m 1 + +m +n 1 + +n +m 1+ +m m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 e km 1+ +m + i..33 Cse 3. When α =1,β =. The function f z ecomesofthefollowingform: f z= + + πs z θ + πs θ 1 From.3weesilotin f z= + + πs z θ + πs s=1 + + z θ + + πs z θ + πs s= z θ f z + π= Compring.35nd.36, when is even, we hve..3, f z + π=f z..37 B.5nd.8, nd noting tht + = k,weotin f z + π= + + πs + nπ z θ + πs + nπ θ 1 = 1 q 1 e iz When nd re even, then is lso even, we hve θ πs z θ + πs..38 f z + π=q 1 e iz f z..39 We construct the function f z θ 3,.37nd.39, we find tht the function f z n elliptic function with doule periods π nd π nd onl hs simple pole t z = π + π z θ 3 z is

8 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 8 of 15 in the period prllelogrm. Hence the function denoted C 1, ;,, i.e., we hve f z θ 3 z = C 1, ;,, f z θ 3 is constnt, s, this constnt is z f z=c 1, ;, θ 3 z..0 B 1.3,.3nd.0, we hve + + πs = C 1, ;, B 1.1nd1., we otin i q m 1,...,m,n 1,...,n = e i m 1 + +m+ = C 1, ;, θ + πs q m e miz..1 m 1 + +m +n 1 + +n +m 1 + +m q e izm 1 + +m+n 1 + +n + n 1 + +n e iπsm 1 + +m+n 1 + +n + q m e miz.. Bequtingtheconstnt termofoth sides of., we otin C 1, ;, =i q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 m 1+ +n m 1 + +m +n 1 + +n +m 1 + +m q e i [km 1+ +m + ]..3 When is even, n nd re odd, then is lso odd, we hve f z + π= q 1 e iz f z.. We construct the function f z θ.b.37nd., we find tht the function f z z θ z is n elliptic function with doule periods π nd π nd onl hs simple pole t z = π + π in the period prllelogrm. Hence the function f z θ is constnt, s, this constnt is z denoted C 1, ;,, i.e., we hve f z θ z = C 1, ;,, f z=c 1, ;, θ z..5

9 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 9 of 15 B 1.,.3 nd.5, we hve + + πs = C 1, ;, θ + πs 1 m q m e miz..6 B 1.1nd1., we otin i q m 1,...,m,n 1,...,n = e i m 1 + +m+ = C 1, ;, m 1 + +m +n 1 + +n +m 1 + +m q e izm 1 + +m+n 1 + +n + n 1 + +n e iπsm 1 + +m+n 1 + +n + 1 m q m e miz..7 B equting the constnt term of oth sides of.7, we otin C 1, ;, =i q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 m 1+ +n m 1 + +m +n 1 + +n +m 1 + +m q e i [km 1+ +m + ]..8 Clerl, in.3nd.8, we hve where C 1, ;, =C 1, ; C 1, ;, =i q e i,, m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 m 1+ +n q m 1 + +m +n 1 + +n +m 1+ +m e km 1+ +m + i..9 InthesmemnnersinCse3,wecnotinCseelow. Cse. When α =,β =. The function f z ecomesofthefollowingform: f z= + + πs z θ + πs θ When nd re even, we hve..50 f z=c, ;, θ 3 z,.51

10 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 10 of 15 C, ;, =q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 n 1+ +n m 1 + +m +n 1 + +n +m 1 + +m q e i [km 1+ +m + ]..5 When is even, n nd re odd, we hve f z=c, ;, θ z,.53 C, ;, =q e i m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 n 1+ +n m 1 + +m +n 1 + +n +m 1 + +m q e i [km 1+ +m + ]..5 Clerl, in.5nd.5, we hve where C, ;, =C, ; C, ;, =q e i,, m 1,...,m,n 1,...,n = m 1 + +m +n 1 + +n +=0 1 n 1+ +n q m 1 + +m +n 1 + +n +m 1+ +m e km 1+ +m + i..55 Therefore we complete the proof of Theorem 1. 3 SomespecilcsesofTheorem1 In this section we give some specil cses of Theorem 1 nd otin some interesting identities of thet functions. Corollr 1 For n positive integer n, we hve n 1 n + + πs nn n 1 = θ n + + πs nn =ne 3i/ θ n 3 0 n = C 13, ;, n n + πs nn θ 3.1 n + πs n 3. n θ 3 z 3.3 θ 3 z =C, ;, n θ 3 z, 3. where C 13, ;, nd C, ;, re defined.1 nd.55, respectivel.

11 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 11 of 15 Proof Tking = =ndα =1,β =3in1.11, we hve C 13, ;, = q 1 = nq 1 e 5i = nq 1 e 5i m 1,m,n 1,n = m 1 +m +n 1 +n +1=0 m,n,l= m,n,l= 1 m 1+m q m 1 +m +n 1 +n +m 1+m e 8im 1+m e 5i 1 l q m +n +l m +l+n+1 +l e 8il 1 l+m q m +m+l +l+n +l+n+m+1 e 8il+m =ne 3i θ Tking = =ndα =,β =in1.11, we hve C, ;, =nq 1 m 1,m,n 1,n = m 1 +m +n 1 +n +1=0 1 n 1+n q m 1 +m +n 1 +n +m 1+m e 8im 1+m e 5i =ne 3i θ Oviousl, we find tht C 13, ;, =C, ;, =ne 3i θ 1 θ 3 0. Tking n = 1 nd letting z z,, in Corollr 1, we get the following identities for thet functions: z + θ 3 z +θ 1 z + + π + z + + π z + π + z + + 3π z + 3π = θ z + θ z +θ z + + π + θ z + + π θ z + π + θ z + + 3π θ z + 3π z + π θ z + π =e 3i/ θ 1 θ 3 0 θ 3z. 3.9 Further, tking = 0 in the ove identities, we otin the following dditive formuls of the thet function θ 3 z : 1 θ 3 z = θ 3 + z + π [ z θ 3 z +θ 1 z + π z + π + z + 3π z + π z + 3π ] 3.10

12 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 1 of 15 1 = θ 3 + θ z + π [ θ z θ z +θ θ z + π z + π θ + θ z + 3π z + π θ z + 3π ], 3.11 where θ 1 = θ 1 0, θ 3 = θ 3 0. Similrl, we hve the following. Corollr For n positive integer n, we hve n 1 n + + πs nn θ n 1 = θ n + + πs nn =ne 3i/ θ θ n 3 0 n = C 1, ;, n n + πs nn θ 3 z =C 3, ;, n 3.1 n + πs n 3.13 n θ 3 z 3.1 θ 3 z, 3.15 where C 1, ;, =C 3, ;, =ne 3i θ θ 3 0 re defined.9 nd.33, respectivel. Tking n = 1 nd letting z z,, in Corollr, we get the following identities of thet functions: z + θ z +θ 1 z + + π θ + z + + π θ z + π + z + + 3π θ z + 3π = θ z + θ 3 z +θ z + + π + θ z + + π z + π + θ z + + 3π z + 3π z + π z + π =e 3i/ θ θ 3 0 θ 3z Further, tking = 0 in the ove identities, we otin other dditive formuls for the thet function θ 3 z s follows: 1 θ 3 z = θ θ 3 + z + π [ z θ z +θ 1 θ z + π z + π θ + z + 3π z + π θ z + 3π ] 3.19

13 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 13 of 15 1 = θ θ 3 + θ z + π [ θ z θ 3 z +θ z + π z + π + θ z + 3π z + π z + 3π ], 3.0 where θ = θ 0, θ 3 = θ 3 0. Tking =, =1in1.9, 1.10nd1.11, we hve the following. Corollr 3 For n odd, we hve 3n 1 3n + + πs z 3n θ 3n 3n + πs 3n3n =3nθ θ 3n 3n = C 1, 1;, 3n θ z 3.1 θ z, 3. 3n 1 θ 3n + + πs z 3n θ 3n 3n + πs 3n3n =3nθ θ 3n 1 3n = C, 1;, 3n θ z 3.3 θ z, 3. where C 1, ;, nd C, ;, re defined.9 nd.55, respectivel. Proof Tking =, =1ndα = 1,, β =in1.11, we hve C 1, 1;, = 3nq 1 e 6i =3nq 1 e 6i =3nq 1 m 1,m,n 1 = m 1 +m +n 1 +1=0 l, l, C, 1;, =3nq 1 e 6i 1 m 1+m +n 1 q m 1 +m +n 1 +m 1+m e 6m 1+m i q m +l m +l+1 +l e 6il q m +m+l +l+l+m+1 e 3il+m+ =3nθ 3 0 θ 3, 3.5 =3nq 1 e 6i = 3nq 1 m 1,m,n 1 = m 1 +m +n 1 +1=0 l, l, 1 n 1 q m 1 +m +n 1 +m 1+m e 6m 1+m i 1 l+1 q m +l m +l+1 +l e 6il 1 l+m q m +m+l +l+l+m+1 e 3il+m+ =3nθ

14 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 1 of 15 Tking n = 1 nd letting z 3z,, 3 in Corollr 3, wegetthefollowing identities for thet functions: z + θ z + z + + π θ z + π z + + π θ z + π =3θ 3 0 θ θ z 3.7 nd θ z + θ z +θ z + + π θ z + π θ z + + π θ z + π =3θ θ z. 3.8 Further tking = 0 in the ove identities, we otin the following dditive formuls for the thet function θ z s follows: 1 θ 3z 3= 3θ 3 θ + θ 1 [ z θ z + z + π 3 θ z + π 3 z + π θ z + π 3 3 ] 3.9 nd 1 θ 3z 3= 3θ 3 + θ [ θ z θ z +θ z + π 3 θ z + π 3 z + π θ z + π 3 3 ], 3.30 where θ 1 = θ 1 0, θ = θ 0, θ 3 = θ 3 0. Corollr When n is odd, we hve 1 s θ k z + πs = C 33 k,0;0,θ z, 3.31 where C 33 k,0;0, is defined 1.7. Proof Replcing z z + π + π nd in 1.6, we otin z + π + n π + πs = C 33 k,0;0,θ 3 z + π + π k. 3.3 Sustituting n n + 1 in the left-hnd side of 3.3, we get 3.31.

15 Ci et l. Boundr Vlue Prolems 013, 013:59 Pge 15 of 15 Competing interests The uthors declre tht the hve no competing interests. Authors contriutions All uthors contriuted equll in writing this pper, nd red nd pproved the finl mnuscript nd. Acowledgements Dedicted to Professor Hri M Srivstv. The present investigtion ws supported, in prt, Reserch Project of Science nd Technolog of Chongqing Eduction Commission, Chin under Grnt KJ1065, Fund of Chongqing Norml Universit, Chin under Grnt 10XLR017 nd 011XLZ07 nd Ntionl Nturl Science Foundtion of Chin under Grnt nd Received: 10 Decemer 01 Accepted: 6 Ferur 013 Pulished: 6 Mrch 013 References 1. Chn, HH, Liu, ZG, Ng, ST: Circulr summtion of thet functions in Rmnujn s lost noteook. J. Mth. Anl. Appl. 316, Zeng, XF: A generlized circulr summtion of thet function nd its ppliction. J. Mth. Anl. Appl. 356, Liu, ZG: Some inverse reltions nd thet function identities. Int. J. Numer Theor 8, Berndt, BC: Rmnujn s Noteooks, Prt III. Springer, New York Berndt, BC: Rmnujn s Noteooks, Prt V. Springer, New York Chn, SH, Liu, ZG: On new circulr summtion of thet functions. J. Numer Theor 130, Chu, KS: The root lttice A n nd Rmnujn s circulr summtion of thet functions. Proc. Am. Mth. Soc. 130, Chu, KS: Circulr summtion of the 13th powers of Rmnujn s thet function. Rmnujn J. 5, Liu, ZG: A thet function identit nd its implictions. Trns. Am. Mth. Soc. 357, Ono, K: On the circulr summtion of the eleventh powers of Rmnujn s thet function. J. Numer Theor 76, Rmnujn, S: The Lost Noteook nd Other Unpulished Ppers. Nros, New Delhi Shen, LC: On the dditive formul of the thet functions nd collection of Lmert series pertining to the modulr equtions of degree 5. Trns. Am. Mth. Soc. 35, Son, SH: Circulr summtion of thet functions in Rmnujn s lost noteook. Rmnujn J. 8, Zhu, JM: An lternte circulr summtion formul of thet functions nd its pplictions. Appl. Anl. Discrete Mth. 6, Zhu, JM: A note on generlized circulr summtion formul of thet functions. J. Numer Theor 13, doi: / Cite this rticle s: Ci et l.: Some circulr summtion formuls for thet functions. Boundr Vlue Prolems :59.