1, if k = 0. . (1 _ qn) (1 _ qn-1) (1 _ qn+1-k) ... _: =----_...:... q--->1 (1- q) (1 - q 2 ) (1- qk) - -- n! k!(n- k)! n """"' n. k.

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1 Abstract. We prove the ifiite q-biomial theorem as a cosequece of the fiite q-biomial theorem. 1. THE FINITE q-binomial THEOREM Let x ad q be complex umbers, (they ca be thought of as real umbers if the reader prefers,) ad for the momet, q =/:- 1. The fiite q-biomial theorem is, (1) (1 + x)(1 + qx)... (1 + q-1x) = t (q) qk(k-1)/2xk, (q)k(q)-k where is a positive iteger ad (2) (q)k := { (1- q)(1- q 2 )... (1- qk), if k = 1, 2, 3,..., 1, if k = 0. 0 bserve that l. (q) 1m -:-:---=.:...,.---- q-->1 (q)k(q)-k. (1 _ q) (1 _ q-1) (1 _ q+1-k) hm... _: =----_...:... q--->1 (1- q) (1 - q 2 ) (1- qk) -1 -k k! k!(- k)! Therefore idetity (1) reduces to the well-kow biomial theorem i the case q ~ 1. ( ) """"'. k 1 +x = L k!(- k)!x I 2. A PROOF OF THE FINITE q-binomial THEOREM I this sectio, we reproduce the proof of the fiite q-biomial theorem give by G. Polya ad G.L. Alexaderso i their very iterestig paper [4]. Deote the left had side of (1) 02 I Mathematical Medley I Volume 33 No.2 December 2006

2 The q-biomial Theorem I ] 1\ll_ ~ ~~~0~~~~1~ by f(x) ad observe that (3) (1 + x)f(qx) = f(x)(1 + qx). If we write the (3) shows that f(x) = L Qkxk, (1 + x) L Qkqkxk = (1 + qx) L Qkxk. Comparig coefficiets of xk, we fid that, for k 2: 1, Qkqk + Qk-1qk- 1 = Qk + qqk-1, or Sice Q 0 = 1, we coclude that q -k+1 _ 1 k-1 Q k = Q k-1 k q. q -1 Q k - _ (q) k(k-1)/2 q (q)k(q)-k. 3. TWO IDENTITIES OF EULER From ow o we will assume that lql < 1. Lettig oo i the fiite q-biomial theorem (1) ad applyig Taery's theorem [5, 49] to justify lettig oo uder the summatio sig, we deduce a idetity of Euler: (4) g (1 + qkx) = {; q (q)k <Xl <Xl k(k-1)/2 If we set x = -1 i the fiite q-biomial theorem (1) ad multiply by (-l) j (q) we fid that xk. (5) t 1 qk(k-1) /2(-l)-k = O ( q)k( q)-k for = 1, 2, 3,... For = 0, it is clear that the expressio o the left side of (5) is equal to 1. It is kow that if the C(x) = A(x)B(x)... Mathematical Medley I Volume 33 No.2 December 2006 I 03

3 I m Th;;:&iomial Theorem I I I 0~ where Cosequetly, (5) implies that where A(x)B(x) = 1 00 qk(k-1)/2 k A(x) = L () x q k ad ~ k xk B(x) = L.)-1) -(). q k This, together with ( 4), implies aother idetity of Euler: (6) Equatio (4) is valid for all complex umbers x, while equatio (6) is valid for lxl < 1. I each case Jql < THE INFINITE q-binomial THEOREM Replace x with -a i the fiite q-biomial theorem t (1) ad write the result i the form (1 _ a)(1 _ qa)... (1 _ q-la) = qk(k-l)/2(-a)k (q) (q)k(q)-k By applyig the techique i the previous sectio, we fid that f (1- a)(1- qa)... (1- qk-la) xk = (f qk(k-l)/2 ( -ax)k) (f _ _). (q)k (q)k (q)k By Euler's idetities (4) ad (6), we deduce (?) f (1- a)(1- qa) (1- qk-la) xk =IT (1- qkax). (q)k (1 - qkx) This is called the ifiite q-biomial theorem. It is valid for Jxl < 1, Jql < 1 ad ay complex umber a. If we replace a with qa ad let q i the ifiite q-biomial theorem, the we formally obtai the biomial series ~ (a)(a + 1) (a + k- 1) k = ( 1 _ )-a ~ k! X X. 04 I Mathematical Medley I Volume 33 No. 2 December 2006

4 The q-biomial Theorem A rigorous justificatio of the limit process is give i [2, pp ]. The fiite q-biomial theorem (1) is a special case of the ifiite q-biomial theorem (7). To see this, let be a positive iteger, let a = q- ad replace x with -qx i (7). The result is (1). The Euler idetities (4) ad (6) are also special cases of the ifiite q-biomial theorem ( 7). I this sectio we have see that the ifiite q-biomial theorem (7) is i fact a cosequece of the fiite q-biomial theorem (1). 5. THE JACOBI TRIPLE PRODUCT IDENTITY I Sectio 3 we saw that Euler's idetity (4) is a limitig case of the fiite q-biomial theorem (1). There is aother importat limitig case of the fiite q-biomial theorem, which we shall ow obtai. I the fiite q-biomial theorem (1), replace with 2 ad q with q 2, ad the let x = q 1-2 z, to get 2 ( 2) (1 + ql-2z)... (1 + q-lz)(1 + qz)... (1 + q2-lz) = L 2 q 2 2 qk 2-2kzk. (q )k(q h-k Multiply both sides by (q 2 )q 2 z- ad set k = + j i the sum, to get g(l + q 2 Hz)(l + q 2 Hz- 1 )(1- q 2 ;) ~ i; (q~~?:;(~''j:_; qi' z; Applyig Taery's theorem [5, 49], we take the limit as oo. We obtai j = l j = -oo This result is called the Jacobi triple product idetity. It is valid for lql < 1 ad ay o-zero complex umber z. The right had side should be viewed as a Lauret series i the aulus 0 < izl < oo. There is a essetial sigularity at z = 0. The left had side shows that there are zeros at z =..., -q- 3, -q- 1, -q, -q 3,..., ad these are the oly zeros. The above proof of the Jacobi triple product ca be foud i [2, p.497]. A completely differet proof of the Jacobi triple product idetity was give idepedetly by G.E. Adrews [1] ad P.K. Meo [3]. Adrews ad Meo showed that the Jacobi triple product idetity may be proved usig the two Euler idetities (4) ad (6). Sice we have show that the Euler idetities are cosequeces of the fiite q-biomial theorem (1), the proof of Adrews ad Meo gives aother proof of the Jacobi triple product idetity which depeds oly o the fiite q-biomial theorem. Mathematical Medley I Volume 33 No. 2 December 2006 I 05

5 The q-biomial Theorem 6. SUMMARY I this article we have see that the two Euler idetities (4) ad (6), the ifiite q-biomial theorem (7) ad the Jacobi triple product idetity are all cosequeces of the fiite q-biomial theorem ( 1). Historical iformatio about the origis of these idetities is give i [2, pp. 491, 497 ad 501]. Applicatios of the idetities are wide ad varied. For example, see [2, Chapters 10 ad 11]. Refereces [1] G.E. Adrews, A simple proof of Jacobi's triple product idetity, Proc. Amer. Math. Soc., 16 (1965), [2] G. E. Adrews, A. Askey ad R. Roy, Special Fuctios, Cambridge Uiversity Press, Cambridge, [3] P.K. Meo, O Ramauja's cotiued fractio ad related idetities, J. Lodo Math. Soc. 40 (1965), [4] G. Polya ad G.L. Alexaderso, Gaussia biomial coefficiets, Elem. Math., 26 (1971 ), [5] T.J.I.'a Bromwich, A Itroductio to the Theory of Ifiite Series, 2d revised ed (Macmilla, Lodo, 1964). 06 I Mat:hemat:ical Medley I Volume 33 No. 2 December 2006