Q-analogue of summation formulas

introduction

• 초기하 급수의 합공식
  
• q-Chu-Vandermonde
  \(_2\phi_1(q^{-n},b;c;q,q)=\frac{(c/b;q)_n}{(c;q)_n}b^n\)
  
• [GR2004] (1.5.1) Heine's q-analogue of Gauss' summation formula
  \(_2\phi_1(a,b;c,q,c/ab)=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}\) or 
  \(\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}}{(c ,q)_{n}(q ,q)_{n}}(\frac{c}{ab})^{n}=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}\)
  
• [GR2004] (1.7.2) q-analogue of Pfaff-Saalschutz's summation formula
  \(_3\phi_2(a,b,q^{-n};c,abc^{-1}q^{1-n};q,q)=\frac{(c/a,c/b;q)_{n}}{(c,c/ab;q)_{n}}\) or
  \(\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}(q^{-n},q)_{n}}{(c)_{n}(abc^{-1}q^{1-n} ,q)_{n}(q ,q)_{n}}q^{n}=\frac{(c/a,c/b;q)_{n}}{(c,c/ab;q)_{n}}\)
  
• q-analogue of Whipple's theorem
  
• Jackson's q-analogue of Dougall's theorem
  

 

books

• [GR2004] Gasper, George; Rahman, Mizan Basic hypergeometric series 2004