# Bailey–Daum q-Kummer sum

(0.004 seconds)

## 1—10 of 373 matching pages

โบ
โบ
โบ
โบ
โบ
โบ
โบ
โบ
โบ
##### 4: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
โบ
17.8.1 $\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};$
โบ
17.8.3 $\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.$
โบ
โบ
##### 5: 4.48 Software
โบ
• Bailey (1993). Fortran.

• โบSee also Bailey (1995), Hull and Abrham (1986), Xu and Li (1994). …
##### 6: 17.1 Special Notation
โบ
$f(\chi_{1};\chi_{2},\dots,\chi_{n})+\operatorname{idem}\left(\chi_{1};\chi_{2}% ,\dots,\chi_{n}\right)=\sum_{j=1}^{n}f(\chi_{j};\chi_{1},\chi_{2},\dots,\chi_{% j-1},\chi_{j+1},\dots,\chi_{n}).$
โบA slightly different notation is that in Bailey (1964) and Slater (1966); see §17.4(i). …
##### 7: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions
โบ
###### Bailey’s ${{}_{2}\psi_{2}}$ Transformations
โบ
17.10.4 ${{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.$
โบ
โบ
โบ
โบ
โบ
##### 9: 16.12 Products
โบ
16.12.3 $\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},$ $|z|<1$.
##### 10: Bibliography B
โบ
• D. H. Bailey (1993) Algorithm 719: Multiprecision translation and execution of Fortran programs. ACM Trans. Math. Software 19 (3), pp. 288–319.
• โบ
• D. H. Bailey (1995) A Fortran-90 based multiprecision system. ACM Trans. Math. Software 21 (4), pp. 379–387.
• โบ
• W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
• โบ
• W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
• โบ
• W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.