q GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS

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1 q GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS BRUCE C. BERNDT 1 and AE JA YEE 1. Introduction Recall that the q-gauss summation theorem is given by (a; q) n (b; q) ( n c ) n (c/a; q) (c/b; q) =, (1.1) (c; q) n (q; q) n ab (c; q) (c/(ab); q) n=0 where c/(ab) < 1. Gauss s name is attached to this theorem, because it is the q- analogue of Gauss s summation for ordinary or Gaussian hypergeometric series. The theorem (1.1) was however first discovered by E. Heine [8] in We only know of two proofs of (1.1) up to recent times. The first proof, due to Heine [8], uses what we now call Heine s transformation, and this proof can be found in the texts of G. E. Andrews [1, p. 20], Andrews, R. Askey, and R. Roy [2, p. 522], and G. Gasper and M. Rahman [7, p. 10]. The second proof employs the q-analogue of Saalschütz s theorem and can be read in the texts of W. N. Bailey [3, p. 68] and L. J. Slater [10, p. 97]. The purpose of this short note is to present two different approaches to the proof of the q-gauss summation theorem. The first proof is due to Ramanujan and is published in a fragment with his lost notebook [9, pp ]. Ramanujan s proof, which encompasses Corollary??, Lemma 2.3, and Theorem 2.4 in Section 2, does not appear to have been noticed by many mathematicians. The second approach uses the theory of partitions and rests upon a combinatorial interpretation of each side of (1.1). Combinatorial proofs of (1.1) have been given by S. Corteel and J. Lovejoy [6], Corteel [5], and the second author [11]. The new concept of an overpartition was employed in the latter three papers. Here, without the use of overpartitions, a variation of the second author s proof [11] is presented. 2. Ramanujan s Proof of the q Gauss Summation Theorem Ramanujan gives a brief sketch of his proof of the q-analogue of Gauss s summation theorem on pages 268 and 269 of [9]. We now reconstruct and (mildly) correct his proof. Recall the q-binomial theorem [4, p. 14, Entry 2]. 1 Research partially supported by grant MDA from the National Security Agency. 1

2 2 BRUCE C. BERNDT and AE JA YEE Lemma 2.1. If a < 1, then ( b) (a) = ( b/a) k a k (q) k. (2.1) Corollary 2.2. If n is any nonnegative integer, then (a) n = ( 1) k (qn+1 k ) k q k(k 1)/2 a k. (2.2) (q) k Proof. Apply (2.1) with a replaced by aq n, let b = a, and use elementary manipulation. Lemma 2.3. If c 0 and n is any nonnegative integer, then c n c j (1/c) j (q n+1 j ) j =. (2.3) (q) j Proof. Denote the right side of (2.4) by g(c) and apply (2.2) with a = 1/c and n = j in the definition of g(c) to find that g(c) = j The coefficient of c r, 0 r n, above is ( 1) k (qj+1 k ) k (q n+1 j ) j (q) j (q) k q k(k 1)/2 c j k =: Now we can easily verify that and a r c r. n r a r = ( 1) k (qr+1 ) k (q n+1 r k ) r+k q k(k 1)/2. (2.4) (q) r+k (q) k (q r+1 ) k (q) r+k = 1 (q) r (q n+1 r k ) r+k = (q n+1 r k ) k (q n+1 r ) r. Using these last two equalities in (2.5), we find that a r = (qn+1 r ) r (q) r n r ( 1) k (qn r+1 k ) k (q) k q k(k 1)/2 = (qn+1 r ) r (q) r (1) n r = by (2.2). This therefore completes our proof of Lemma 2.3. r=0 { 1, if r = n, 0, otherwise. Theorem 2.4 (q Gauss Summation Theorem). If abc < 1 and bc 0, then (ac) (abc) = (a) (ab) n=0 (1/b) n (1/c) n (a) n (q) n (abc) n. (2.5)

3 q GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS 3 Proof. We rewrite the right side of (2.6) in the form (aq j ) (1/b) j (1/c) j (abc) j (2.6) (ab) (q) j and examine the coefficient of a n, n 0, on each side of (2.6). From (2.1), with a replaced by ab and b replaced by aq j, we find that (aq j ) (q j /b) k = (ab) k. (2.7) (ab) (q) k The coefficient of a n j in (2.8) is (q j /b) n j (q) n j b n j, and so the coefficient of a n in (2.7) equals (1/b) j (1/c) j (q j /b) n j b n c j = (1/b) nb n (q) j (q) n j (q) n c j (1/c) j (q n+1 j ) j (q) j = (1/b) nb n (q) n c n, (2.8) by Lemma 2.3. But, by (2.1), with a replaced by abc and b replaced by ac, (ac) (1/b) n = (abc) n. (2.9) (abc) (q) n n=0 So, the coefficient of a n in (2.10) is precisely that on the right side of (2.9). Hence, (2.6) immediately follows, since the coefficients of a n, n 0, on both sides of (2.6) are equal. The proof of Theorem 2.4 is therefore complete. 3. Combinatorics in the q-gauss summation Let P n be the set of partitions into nonnegative parts less than n, D n be the set of partitions into distinct nonnegative parts less than n, and P n + be the set of partitions into positive parts less than n. Let P and D be the set of partitions into nonnegative parts and nonnegative distinct parts, respectively. It is well known that 1/(x; q) n and ( y; q) n are the generating functions for P n and D n, respectively, where the exponents of x and y are equal to the numbers of parts of partitions, and 1/(x; q) and ( y; q) are the generating functions for P and D, respectively (see [1]). Thus, by examining both sides of (1.1), we obtain a partition theoretic interpretation of the q-gauss summation, which is stated in the following theorem. Theorem 3.1. There is a one to one correspondence between n=0d n P n D n P + n+1 and D P D P. We need the following lemmas to prove Theorem 3.1. Let P(n) be the set of partitions into n nonnegative parts and D(n) be the set of partitions into n distinct nonnegative parts.

4 4 BRUCE C. BERNDT and AE JA YEE Lemma 3.2. There is a one to one correspondence between D n P(n) and n D(k) P(n k) such that (π 1, π 2 ) corresponds to (π 3, π 4 ) D(k) P(n k) for any partition π 1 D n into k parts. Proof. In this lemma, we put the parts of a partition in nondecreasing order. Let (π 1, π 2 ) D n P(n) be given. Let k be the number of the parts of π 1. Define π 3 by π 3 i = π 2 π 1 i +1 + π1 i for 1 i k, and define π 4 as the partition of the remaining (n k) parts of π 2. Since π 1 is a partition into k distinct parts for k n, π 3 is a partition in D(k). Meanwhile, π 4 is a partition in P(n k). The process is invertible, since we can find the right positions for the parts of π 3 when we combine π 3 and π 4 by the fact that there is a unique j such that c j m j c j+1 for any nonnegative integer m and any nondecreasing array (c 1,..., c s ), where c 0 = and c s+1 =. We describe the reverse process. Let (π 3, π 4 ) D(k) P(n k) be given. We begin with π 4, and progressively build partitions using the parts of π 3 from the largest one until all parts of π 3 have been used. We define a map φ that will be employed at each step. Let ρ be a partition with l parts. Then for any integer m, we can find the unique j such that ρ j m j ρ j+1, where ρ 0 = and ρ l+1 =. Define φ as φ(ρ, m) = (ρ, j), where ρ i, if 1 i j, ρ i = m j, if i = j + 1, ρ i 1, if i > j + 1. We successively apply φ a total of k times as follows: φ(π 4(i), π 3 i ) = (π 4(i+1), π 1 i ), 1 i k, where π 4(1) = π 4. Let π 2 = π 4(k+1) and π 1 be the partition consisting of integers πi 1 obtained at each step. Let P(k, m) be the set of partitions into k parts m. Lemma 3.3. There is a one to one correspondence between D n D(n) and n D(n k) P(k, k 1) such that (µ 1, µ 2 ) corresponds to (µ 3, µ 4 ) D(n k) P(k, k 1) for any partition µ 1 D n into k parts. Proof. Let (µ 1, µ 2 ) D n D(n) be given. Let k be the number of parts of µ 1. Define µ 4 as µ 4 i = µ 2 µ 1 i +1 + µ1 i for 1 i k and define µ 3 as the partition of the remaining (n k) parts of µ 2. Note that µ 4 is a partition in P(k), since µ 1 is a partition into k distinct parts for k n, µ 2 is a partition

5 q GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS 5 into distinct parts, and in the process, µ 1 i is added to the (µ 1 i + 1) st smallest part of µ 2. Thus µ 4 is in P(k, k 1). Meanwhile, µ 3 is a partition in D(n k). We can show the process is reversible as we saw in the proof of Lemma 3.2, for the inverse process can be defined similarly to that described in the proof of Lemma 3.2. We omit the proof of the inverse process. Lemma 3.4. There is a one to one correspondence between D n P + n+1 and n D(k) P(n k). Proof. Taking the conjugate of a partition in P n+1 + gives a partition in P(n). Thus by Lemma 3.2, we see the statement in this lemma holds. We now prove Theorem 3.1. Proof. For any n 0, let (σ 1, σ 2, σ 3, σ 4 ) be in D n P n D n P n+1. + We take the bijection described in Lemma 3.4 for σ 3 and σ 4. Let σ 3 and σ 4 be the resulting partitions. Note that σ 3 D(k) and σ 4 P(n k). Let σ 1 be the partition of the parts of σ 1 less than (n k), and σ 1 be the partition of the remaining parts of σ 1. We apply the map defined in Lemma 3.2 to ( σ 1, σ 4 ). Let the resulting partitions be (ν 3, ν 4 ). We subtract (n k) from each part of σ 1 and call the resulting partition σ. We apply the map defined in Lemma 3.3 to ( σ, σ 3 ). Let the resulting partitions be (ν 1, ν 2 ). Let ν 2 be the partition obtained by adding (n k) to each part of ν 2 and adding the parts of σ 2 as parts. Therefore we obtain a quadruple (ν 1, ν 2, ν 3, ν 4 ) in D P D P. The process above is invertible. Let (ν 1, ν 2, ν 3, ν 4 ) be given. Let M be the sum of the numbers of parts of ν 1, ν 3, and ν 4. Take the maximum n (n M) rectangle that fits inside the Ferrers graph of ν 2. Let σ 2 be the partition consisting of the parts to the right of the n (n M) rectangle. In the process above, we added σ 2 to ν 2 whose (n M) parts are greater than or equal to n. Thus taking the maximum rectangle gives us back σ 2. Meanwhile, we applied the bijections defined in Lemmas 3.2 to 3.4. Thus we see the the process is invertible. Note that the numbers of the parts of π 1 and µ 1 are equal to the numbers of parts of π 3 and µ 4 in Lemmas 3.2 and 3.3, respectively. Thus the exponents of a, b and c in (1.1) are handled correctly in Theorem 3.1. The proof of Theorem 3.1 has a graphical representation as in Figure 1, where n = 5, the dots below the diagonal denote partitions obtained by applying to (σ 3, σ 4 ) the process in the proof of Lemma 3.4, the dots above the diagonal with denote the parts of σ 1, and the dots to the right of the vertical dashed line denote the parts of σ 2. To obtain (ν 1, ν 2, ν 3, ν 4 ), we read the dots in columns such that the columns with only above the diagonal become the parts of ν 3, the columns without become the parts of ν 4, the columns with only below the diagonal become the parts of ν 1, and the columns with below and above the diagonal and the columns to the right of the vertical dashed line become the parts of ν 2. References [1] G. E. Andrews, The Theory of Partitions, Addison Wesley, Reading, MA, 1976; reissued: Cambridge University Press, Cambridge, [2] G. E. Andrews, R. A. Askey, and R. Roy, Special Functions, University Press, Cambridge, 1999.

6 6 BRUCE C. BERNDT and AE JA YEE * * * * Figure 1. (σ 1, σ 2, σ 3, σ 4 ) = (3 1, , 3 1 0, ) (ν 1, ν 2, ν 3, ν 4 ) = (7 1, , 7, 3, ) [3] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935; reprinted by Stechert-Hafner, New York, [4] B. C. Berndt, Ramanujan s Notebooks, Part III, Springer Verlag, New York, [5] S. Corteel, Particle seas and basic hypergeometric series, Adv. Appl. Math. 31 (2003), [6] S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ramanujan s 1 ψ 1 summation, J. Combin. Thy. Ser. A 97 (2002), [7] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encycl. Math. Applics., Vol. 35, Cambridge University Press, Cambridge, [8] E. Heine, Untersuchungen über die Reihe 1 + (1 qα )(1 q β ) (1 q)(1 q γ ) x + (1 qα )(1 q α+1 )(1 q β )(1 q β+1 ) (1 q)(1 q 2 )(1 q γ )(1 q γ+1 x 2 +, ) J. Reine Angew. Math. 34 (1847), [9] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, [10] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, [11] A. J. Yee, Combinatorial proofs of Ramanujan s 1 ψ 1 summation and the q-gauss summation, J. Combin. Thy. Ser. A, to appear. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA address: berndt@math.uiuc.edu Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA address: yee@math.psu.edu