qBeta函数是B函数的q模拟
![{\displaystyle B_{q}(a,b)={\frac {\Gamma _{q}(a)\cdot \Gamma _{q}(b)}{\Gamma _{q}(a+b)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/062ab8daba56c8a4c729113debdaac8282e79c48)
其中
是QΓ函数。
当
且
时,该函数亦可以q积分和q阶乘幂定义:[1]:1481
![{\displaystyle B_{q}(a,b)=\int _{0}^{1}x^{b-1}{\frac {(qx;q)_{\infty }}{(q^{a}x;q)_{\infty }}}\mathrm {d} _{q}x.}](//wikimedia.org/api/rest_v1/media/math/render/svg/1e1f1e09943aee674d24a7dfa769311c8eb1c423)
其与基本超几何函数
的关系为:[1]:1482
![{\displaystyle B_{q}(a,b)={}_{2}\phi _{1}(q^{1-a},q^{b};q^{b+1};q,q^{a}).}](//wikimedia.org/api/rest_v1/media/math/render/svg/6e4a9e9f3bd01affcd7855eba9086b491ef27b59)
Ahmed Salem. Generalized q-integrals via neutrices: Application to the q-beta function. Filomat (Faculty of Sciences and Mathematics, University of Niš, Serbia). 2013, 27 (8): 1473–1483. doi:10.2298/FIL1308473S.
- Frank Oliver. NIST Handbook of Mathematical Functions. Cambridge University Press. 2010: 145.