کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
5775070 | 1413574 | 2017 | 21 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
The t-coefficient method III: A general series expansion for the product of theta functions with different bases and its applications
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
آنالیز ریاضی
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
By means of Jacobi's triple product identity and the t-coefficient method, we establish a general series expansion formula for the product of arbitrary two theta functions with bases p and q:θ(az;p)θ(bzt;q)=ân=âââÏp(n)(az)nθ((â1/a)tbpt(t+1)/2âtn;pt2q), from which some new results on products of arbitrary finitely many theta functions and theta identities associated with Ramanujan's circular summation can be derived, among them the most interesting ones includeâk=0mnâ1âi=1nθ(âayiÏk;q)=mnÎ(0;y1,y2,â¯,yn)Ãθ(â(y1y2â¯yn)mamnqnm22ânm2;qnm2), where Ï is a primitive mn-th root of unity andÎ(s;y1,y2,â¯,yn)=âi1+i2+â¯+in=sq12âj=1nij2âj=1nyjij. Furthermore, for y1y2y3=âq and Ï=expâ¡(Ïi/3),Î3(0;y1,y2,y3)+qâ3/2Î3(1;y1,y2,y3)=1(q;q)â3âi=02âj=13θ(âyjÏ2i;q). The former contains Chan and Liu's circular summation (cf. Chan and Liu (2010) [12]) as a special case y1y2â¯yn=1. The latter generalizes both Borweins and Garvan's well-known cubic theta function identity and Schultz's bivariate-generalization (cf. Borwein et al. (1994) [9] and Schultz (2013) [28]), in which y1=y2=y3=Ïq1/3 and y1=y3/z1,y2=y3/z2,y33=âqz1z2 respectively.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Mathematical Analysis and Applications - Volume 449, Issue 1, 1 May 2017, Pages 244-264
Journal: Journal of Mathematical Analysis and Applications - Volume 449, Issue 1, 1 May 2017, Pages 244-264
نویسندگان
Xinrong Ma, Ruizhong Wei,