# q-binomial theorem

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##### 2: 17.2 Calculus
###### §17.2(ii) Binomial Coefficients
17.2.27 $\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)% _{m}\left(q;q\right)_{n-m}}\\ =\frac{\left(q^{-n};q\right)_{m}(-1)^{m}q^{nm-\genfrac{(}{)}{0.0pt}{}{m}{2}}}{% \left(q;q\right)_{m}},$
17.2.28 $\lim_{q\to 1}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{(}{)}{0.0pt}{}{n}{m}=% \frac{n!}{m!(n-m)!},$
###### §17.2(iii) BinomialTheorem
When $a=q^{m+1}$, where $m$ is a nonnegative integer, (17.2.37) reduces to the $q$-binomial series …
##### 3: 26.9 Integer Partitions: Restricted Number and Part Size
26.9.4 $\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}=\prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}},$ $n\geq 0$,
is the Gaussian polynomial (or $q$-binomial coefficient); see also §§17.2(i)17.2(ii). …
26.9.5 $\sum_{n=0}^{\infty}p_{k}\left(n\right)q^{n}=\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1% +\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m},$
26.9.6 $\sum_{n=0}^{\infty}p_{k}\left(\leq m,n\right)q^{n}=\genfrac{[}{]}{0.0pt}{}{m+k% }{k}_{q}.$
26.9.7 $\sum_{m,n=0}^{\infty}p_{k}\left(\leq m,n\right)x^{k}q^{n}=1+\sum_{k=1}^{\infty% }\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{q}x^{k}=\prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}.$
##### 4: Bibliography K
• Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series ${}_{r+3}F_{r+2}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
• B. J. King and A. L. Van Buren (1973) A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.
• T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.
• C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
• ##### 5: 17.3 $q$-Elementary and $q$-Special Functions
17.3.8 $A_{m,s}\left(q\right)=q^{\genfrac{(}{)}{0.0pt}{}{s-m}{2}+\genfrac{(}{)}{0.0pt}% {}{s}{2}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}\genfrac{[}{]}% {0.0pt}{}{m+1}{j}_{q}\frac{(1-q^{s-j})^{m}}{(1-q)^{m}}.$
##### 6: 26.16 Multiset Permutations
26.16.1 $\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q% }=\prod_{k=1}^{n-1}\genfrac{[}{]}{0.0pt}{}{a_{k}+a_{k+1}+\cdots+a_{n}}{a_{k}}_% {q},$
##### 7: 26.10 Integer Partitions: Other Restrictions
26.10.3 $(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,\mathcal{D},n\right)x^{m}q^{n}=% \sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}q^{m(m+1)/2}x^{m}=\prod_{j=1}^{% k}(1+x\,q^{j}),$ $|x|<1$,
The Chinese remainder theorem states that a system of congruences $x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_{1}$), (mod $m_{2}$), (mod $m_{3}$), and (mod $m_{4}$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $\pmod{m}$, which is correct to 20 digits. …