2 B. C. BERND T Replacing a by qa, where a 0, and using L'Hospital's rule while letting q tend to 1, we find that lim Mk = lim l qa 1 n a+l n a+n— 1 q-+i (1 - g)n PI 1 - q 1-q (1.1.3) = a ( a + l)---(a + n - l ) . 1 - 9 The expression on the right side above is called a rising or shifted factorial, and so we see that (a\ q)n may be considered as an analogue of the rising factorial. We next define the g-binomial coefficient, or Gaussian coefficient, [^], which is an analogue of the ordinary bino- mial coefficient ( ^ ) . Definition 1.1.2. Let n and m denote integers. Then the Gaussian coefficient is defined by (1.1.4) (l)n if 0 m n, otherwise. Exercise 1.1.3. Using (1.1.3) three times, each time with a = 1, show that lim Thus, the ^-binomial coefficient tends to the ordinary binomial coefficient when q 1. From the definition (1.1.4) it is not obvious that the ^-binomial coefficients are polynomials in q. Exercise 1.1.4. Using the definition (1.1.4), readers should first prove the first q-analogue of Pascal's formula given below. Lemma 1.1.5. For n 1, (1.1.5) n - 1 m 1 + qn n - 1 m Exercise 1.1.6. Second, employing Lemma 1.1.5 and induction on n, prove that [^] is a polynomial in q of degree m(n m).
Previous Page Next Page