高斯二项式系数

高斯二项式系数 (也称作 高斯系数, 高斯多项式, 或 q-二项式系数)在数学里是指二项式系数的q-模拟。 高斯二项式系数被定义为： ( m r ) q = { ( 1 − q m ) ( 1 − q m − 1 ) ⋯ ( 1 − q m − r + 1 ) ( 1 − q ) ( 1 − q 2 ) ⋯ ( 1 − q r ) r ≤ m 0 r > m {\displaystyle {m \choose r}_{q}={\begin{cases}{\frac {(1-q^{m})(1-q^{m-1})\cdots (1-q^{m-r+1})}{(1-q)(1-q^{2})\cdots (1-q^{r})}}&r\leq m\\0&r>m\end{cases}}} 其中， m 和 r 是非负整数。 当 r = 0时值为1。 高斯二项式系数计算一个有限维向量空间的子空间数。令q表示一个有限域里的元素数目，则在q元有限域上n维向量空间的k维子空间数等于 ( n k ) q . {\displaystyle {\binom {n}{k}}_{q}.} ( 0 0 ) q = ( 1 0 ) q = 1 {\displaystyle {0 \choose 0}_{q}={1 \choose 0}_{q}=1} ( 1 1 ) q = 1 − q 1 − q = 1 {\displaystyle {1 \choose 1}_{q}={\frac {1-q}{1-q}}=1} ( 2 1 ) q = 1 − q 2 1 − q = 1 + q {\displaystyle {2 \choose 1}_{q}={\frac {1-q^{2}}{1-q}}=1+q} ( 3 1 ) q = 1 − q 3 1 − q = 1 + q + q 2 {\displaystyle {3 \choose 1}_{q}={\frac {1-q^{3}}{1-q}}=1+q+q^{2}} ( 3 2 ) q = ( 1 − q 3 ) ( 1 − q 2 ) ( 1 − q ) ( 1 − q 2 ) = 1 + q + q 2 {\displaystyle {3 \choose 2}_{q}={\frac {(1-q^{3})(1-q^{2})}{(1-q)(1-q^{2})}}=1+q+q^{2}} ( 4 2 ) q = ( 1 − q 4 ) ( 1 − q 3 ) ( 1 − q ) ( 1 − q 2 ) = ( 1 + q 2 ) ( 1 + q + q 2 ) = 1 + q + 2 q 2 + q 3 + q 4 {\displaystyle {4 \choose 2}_{q}={\frac {(1-q^{4})(1-q^{3})}{(1-q)(1-q^{2})}}=(1+q^{2})(1+q+q^{2})=1+q+2q^{2}+q^{3}+q^{4}} 和普通二项式系数一样, 高斯二项式系数是中心对称的: ( m r ) q = ( m m − r ) q . {\displaystyle {m \choose r}_{q}={m \choose m-r}_{q}.} 特别地, ( m 0 ) q = ( m m ) q = 1 , {\displaystyle {m \choose 0}_{q}={m \choose m}_{q}=1\,,} ( m 1 ) q = ( m m − 1 ) q = 1 − q m 1 − q = 1 + q + ⋯ + q m − 1 m ≥ 1 . {\displaystyle {m \choose 1}_{q}={m \choose m-1}_{q}={\frac {1-q^{m}}{1-q}}=1+q+\cdots +q^{m-1}\quad m\geq 1\,.} 当 q = 1 时，有 ( m r ) 1 = ( m r ) {\displaystyle {m \choose r}_{1}={m \choose r}} Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538 Mukhin, Eugene. Symmetric Polynomials and Partitions (PDF). （原始内容 (PDF)存档于2004-12-10）.  (undated, 2004 or earlier). Ratnadha Kolhatkar, Zeta function of Grassmann Varieties （页面存档备份，存于互联网档案馆） (dated January 26, 2004) 埃里克·韦斯坦因. q-Binomial Coefficient. MathWorld.  Gould, Henry. The bracket function and Fontene-Ward generalized binomial coefficients with application to Fibonomial coefficients. Fibonacci Quarterly. 1969, 7: 23–40. MR 0242691.  Alexanderson, G. L. A Fibonacci analogue of Gaussian binomial coefficients. Fibonacci Quarterly. 1974, 12: 129–132. MR 0354537.  Andrews, George E. Applications of basic hypergeometric functions. SIAM Rev. 1974, 16 (4). JSTOR 2028690. MR 0352557. doi:10.1137/1016081.  Borwein, Peter B. Padé approximants for the q-elementary functions. Construct. Approx. 1988, 4 (1): 391–402. MR 0956175. doi:10.1007/BF02075469.  Konvalina, John. Generalized binomial coefficients and the subset-subspace problem. Adv. Appl. Math. 1998, 21: 228–240. MR 1634713. doi:10.1006/aama.1998.0598.  Di Bucchianico, A. Combinatorics, computer algebra and the Wilcoxon-Mann-Whitney test. J. Stat. Plann. Inf. 1999, 79: 349–364. doi:10.1016/S0378-3758(98)00261-4.  Konvalina, John. A unified interpretation of the Binomial Coefficients, the Stirling numbers, and the Gaussian coefficients. Am. Math. Monthly. 2000, 107 (10): 901–910. JSTOR 2695583. MR 1806919.  Kupershmidt, Boris A. q-Newton binomial: from Euler to Gauss. J. Nonlin. Math. Phys. 2000, 7 (2): 244–262. Bibcode:2000JNMP....7..244K. MR 1763640. arXiv:math/0004187 . doi:10.2991/jnmp.2000.7.2.11.  Cohn, Henry. Projective geometry over F1 and the Gaussian Binomial Coefficients. Am. Math. Monthly. 2004, 111 (6): 487–495. JSTOR 4145067. MR 2076581.  Kim, T. q-Extension of the Euler formula and trigonometric functions. Russ. J. Math. Phys. 2007, 14 (3): –275–278. Bibcode:2007RJMP...14..275K. MR 2341775. doi:10.1134/S1061920807030041.  Kim, T. q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15 (1): 51–57. MR 2390694. doi:10.1134/S1061920808010068.  Corcino, Roberto B. On p,q-binomial coefficients. Integers. 2008, 8: #A29. MR 2425627.  Hmayakyan, Gevorg. Recursive Formula Related To The Mobius Function (PDF). [2013-10-22]. （原始内容存档 (PDF)于2021-05-06）.  (2009).