P. Paule. Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electron. J. Combin., 1, 1994. # R10.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....years, dramatic progress has been made in the subject of computer verification of combinatorial identities. An excellent overview of these achievements can be found in the recent book of Petkovsek, Wilf and Zeilberger [10] More recently these methods have been applied to q series as well [8] [9], 12] However, none of these studies claimed to manufacture a q series expansion for a function given only by its power series expansion. This is in marked contrast with Euler s ancient algorithm for the representation of power series by q products [1; p. 98, Ex. 2] Recently the present authors ....

P. Paule, Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type, Electronic Journal of Combinatorics 1, (1994) R10, 1--9.

....q n b Delta Gamma q 2 Gamma q n c Delta S(n Gamma 3) 0; 3.8) which is definitely not the result we want. Here we have an example (of quite a few) where the q Zeilberger algorithm doesn t find a recurrence equation of minimal order, which was pointed out by Paule and Riese (1997) Paule (1994) introduced the method of creative symmetrizing a generalization of which is given by the following lemma resolving the problem of non minimality in most cases. Lemma 3.1. Riese, 1995) If for some c 2 Z bn X k=an F(n; k) bn X k=an F(n; Gammak Gamma c) and M(n; k) F(n; Gammak ....

Paule, P. (1994). Short and easy computer proofs of the Rogers-Ramanujan Identities and of identities of similar type. The Electronic Journal of Combinatorics, 1.

.... (see [1] for details) Schur proved that both dm and e m satisfy the recurrence (13) c m 2 = c m 1 q m c m ; m 0: Nowadays, this identity is proved automatically by invoking the q WZ algorithm [7] and this leads to the rst purely automatic elementary proof of the Rogers Ramanujan identity [6]. In view of this recurrence, dm and e m are nothing but q analogues of the Fibonacci numbers. It turns out that a generalization of the Cassini identity, namely Fm 1 Fm k Fm k 1 Fm = 1) m F k ; admits a q analogue in terms of e m and dm : 14) dm e m k d m k e m = 1) m q ( m 2 ) X ....

.... d1 = 1 (q; q) 1 X k ( 1) k q k(5k 3) 2 = 1 (q 2 ; q 5 ) 1 (q 3 ; q 5 ) 1 ; e 1 = 1 (q; q) 1 X k ( 1) k q k(5k 1) 2 = 1 (q; q 5 ) 1 (q 4 ; q 5 ) 1 : 4 The in nite products are obtained by Jacobi s triple product identity, which also admits a simple computer proof [6]. 4. A New Identity Discovered by Engel Guessing The identities (10) and (11) can be conjectured by Engel guessing after rst multiplying the product by 1 q. An Engel proof is also available [2] using the Santos polynomials de ned by Sm = X j q 4j 2 j m b m 1 4j 2 c ; Tm = X ....

Paule (Peter). { Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electronic Journal of Combinatorics, vol. 1, 1994. { Research Paper 10, 9 pp.

.... k q (5k 2 Gammak) 2 (q; q) n Gammak (q; q) n k 1 q k 2 : Now, the q version of Zeilberger s algorithm applied to this last sum returns the operator Gamma 1 Gamma q n 2 Delta S 2 n Gamma Gamma 1 q Gamma q n 2 q 2n 3 Delta S n q; which is of the desired order [8]. NON MINIMALITY OF ZEILBERGER 15 A consequence of the equality between the sum and its symmetrized version is the nullity n X k= Gamman ( Gamma1) k q (5k 2 Gammak) 2 (q; q) n Gammak (q; q) n k (q k Gamma 1) 0: This suggests q k Gamma 1 as a candidate for Z in the new ....

Paule, P. Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electron. J. Combin. 1 (1994), Research Paper 10, approx. 9 pp. (electronic).

....a second order recurrence equation, whereas the right hand side is annihilated by a recurrence operator of order five. In other words, after running the algorithm we would still have to carry out some steps by hand for proving the equality of these two sums, cf. Ekhad and Tre [5] Recently, Paule [15] found an amazing trick how to boil down q certificates of a large number of applications. This trick is based on the summand s symmetry in k and Gammak. More precisely, assume that the summand F (n; k) satisfies the symmetry condition b X k=a F (n; k) b X k=a F (n; Gammak) Then ....

P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Electronic Journal of Combinatorics, 1 (1994), R10.

....consists in solving a homogeneous system of linear equations with coecients being polynomials in several variables, which is known to be a rather time and especially memory consuming task. Furthermore, Zeilberger s algorithm does not always nd a recurrence of minimal order. While Paule s [9] method of creative symmetrizing overcomes this problem in many instances and, as a side e ect, reduces the run time of the algorithm for certain types of summands, we shall present di erent optimizations in the following. 3 The Method of Uncreative Filtering The rst improvement is based on an ....

P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Electr. J. Combin., 1 (1994), #R10.

....higher than the orders of the minimal recurrences (1 and 0) satisfied by the right sides. They provide many new non trivial examples to the phenomenon described on p. 117 of [PWZ] This phenomenon, of not getting the minimal recurrence, is much more widespread for q series, in which Peter Paule[P] introduced a very useful order reducing preprocessing device. Paule s method is also useful for ordinary hypergeometric sums. At present it is not clear how to apply Paule s method to Andrews s sums, but we suspect that an appropriate generalization will do the job. Perhaps Andrews s identities ....

Peter Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Elect. J. of Combinatorics 1(1994) R10.

....q n b Delta Gamma q 2 Gamma q n c Delta S(n Gamma 3) 0; 3.7) which is definitely not the result we want. Here we have an example (of quite a few) where the q Zeilberger algorithm doesn t find a recurrence equation of minimal order, which was pointed out by Paule and Riese (1995) Paule introduced 1994 the method of creative symmetrizing based on the following lemma which resolves the problem of non minimality in most cases. Lemma 3.1. Riese, 1995) If for some c 2 Z bn X k=an F(n; k) bn X k=an F(n; Gammak Gamma c) Algorithms for q hypergeometric Summation in Computer ....

Paule, P. (1994). Short and easy computer proofs of the Rogers-Ramanujan Identities and of identities of similar type. The Electronic Journal of Combinatorics, 1:1--9.

....this identity, and 56 seconds to find a fifth order operator annihilating the right hand side. From this a proof is easily derived as above. Our generalization of Takayama s algorithm finds the same operators as the general method in 1 second and 23 seconds respectively. It was noted by P. Paule [21] that summing only the even part of the right hand side (i.e. multiplying it by (1 q k ) 2) results in Zeilberger s algorithm finding an operator of order 2 for the right hand side. Using the same trick with our algorithms, we find that Takayama s method benefits from it and yields an operator ....

Paule, P. Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. The Electronic Journal of Combinatorics 1, R10 (July 1994), 1--9.

.... example, several methods are known to deal with this problem (Askey, 1992) Another approach to the automatic proving of this identity consists in proving a finite version of it, which involves ordinary recurrences on the index of summation, whose initial conditions are more easily checked, see (Paule, 1994)) Our elimination algorithms are based on a generalization of the theory of Grobner bases. As Takayama (1989) noticed in the differential difference case, and as was developed by Kandri Rody and Weispfenning (1990) in the more general setting of polynomial rings of solvable type, Buchberger s ....

....to find a fifth order 36 Fr ed eric Chyzak and Bruno Salvy operator annihilating the right hand side. From this a proof is easily derived as above. Our generalization of Takayama s algorithm finds the same operators as the general method in 1 second and 23 seconds respectively. It was noted by Paule (1994) that summing only the even part of the right hand side (i.e. multiplying it by (1 q k ) 2) results in Zeilberger s algorithm finding an operator of order 2 for the right hand side. Using the same trick with our algorithms, we find that Takayama s method benefits from it and yields an ....

Paule, P. (1994). Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. The Electronic Journal of Combinatorics, 1(R10):1--9.

....does not always deliver the recurrence of minimal order. Thus, by looking at the output, we would not be able to realize immediately that the inner sums of S(1) and S(2) have closed forms, because the algorithm does not find the corresponding recurrences of order 1. However, as observed by Paule [2], this phenomenon may be cured in many instances by creative symmetrizing, i.e. by making use of the summand s symmetry. Suppose that ( for some integer linear function t(n, k) Then we have that f(n, f( f( t( f( 1 . f(n, Now, if f(n,t(n, k) f(n, k) ....

....by Peter Paule, Markus Schorn, and Axel Riese ( RISC Linz V 3.30 (03 21 02) n[0] symm fo = 1 (k n j 1 2) n k j 1 2) f . e 0; n[l: gb[symm fo . j 1, k, 0, 2n 1 , n, 1] If 1 2n is a natural number, then: Out[l= 1 2n) SUM(n) 3 2n) SUM(1 n) 0 n[2]: Zb[symm fo . j 2, k, 0, 2n 1 , n, 1] If 1 2n is a natural number, then: Out[12] 1 2n) SUM(n) 5 2n) SUM(1 n) 0 From this and the initial values for n = 1, one gets that the inner sum equals and (2n 1) 2n 1) 2n 1) 2n 3) respectively. This completes the ....

Paule, P.: Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electr. J. Combin. 1 (1994), 4pR10

.... (d m ) and (e m ) satisfy the recurrence c m 2 = c m 1 q c m (m 0) 5) in addition, in the limit m 1 one has d1 = q; q) 1 (6) and e 1 = q; q) 1 : 7) We remark that the proof of the crucial property (5) nowadays can be done automatically with the computer [8]. Also, note that the convertion of the series into the product representation is by Jacobi s triple product identity. See, e.g. 2, p. 21, eq. 2.2.10) We conclude this section by listing the first sequence entries explicitly. Namely, d 0 = 1; d 1 = 0; d 2 = 1; d 3 = 1; d 4 = 1 ....

P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Electron. J. Combin. 1 (1994) R10, 1--9.

....it does not always deliver the recurrence of minimal order. Thus, by looking at the output, we would not be able to realize immediately that the inner sums of S(1) and S(2) have closed forms, because the algorithm does not nd the corresponding recurrences of order 1. However, as observed by Paule [2], this phenomenon may be cured in many instances by creative symmetrizing, i.e. by making use of the summand s symmetry. Suppose that f(n; t(n; k) for some integer linear function t(n; k) Then we have that f(n; k) f(n; t(n; k) f(n; t(n; k) Now, if f(n; t(n; ....

....In the rst part of the proof, we derive a recurrence for f (j; n; k) For this, we start out by utilizing a brand new feature of MultiSum that quickly determines a small structure set. In[1] MultiSum.m MultiSum Package by Kurt Wegschaider c RISC Linz V 1. 45 (03 28 02) In[2]: f = 2n 2 ) 2n 1 k) k 2 (k j n 1=2) In[3] S = FindRecurrence[f , j, 2, fn; kg, f1; 1g, f2; 0g, WZ True, Protocol False, NumericCheck True] Candidate for structure set: Out[3] ff0; 0; 0g; f0; 1; 0g; f0; 1; 1g; f1; 0; 0g; f1; 1; 0g; f1; 1; 1g; f2; 0; 0g; f2; 1; ....

Paule, P.: Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electr. J. Combin. 1 (1994), #R10

.... q q ) SUM[ 2 n] 1 q ) 1 q ) 2 2 n n 2 n 1 2 n 1 3 n (q (1 q q ) 1 q q q ) n 1 2 n SUM[ 1 n] 1 q ) 1 q ) The algorithm qHyper now finds the q hypergeometric solution of this recurrence (after replacing q n by x, and SUM[n k] by Y[qk x] In[10]: SUM[n k. Y[qk x] q(a. n b. xa qb; In[11] qHyper[ Y[x] Warning: irreducible factors of degree 1 in leading coefficient; some solutions may not be found Warning: irreducible factors of degree 1 in trailing coefficient; some solutions may not be found 17 2 2 2 ....

....means that for one solution y n = Y (q n ) we have yn 1 y n = 1 q n 1 ) 1 Gamma q 2n 1 ) From this together with the initial values the right hand side evaluation of (31) is easily computed. Another way to treat the increase of recurrence orders in the q case was found by P. Paule [10]. His method of summing the even part , or variations of it, consists in rewriting the given sum by exploiting symmetries of the summand. After 19 this preprocessing the q version of Zeilberger s algorithm delivers the recursion of minimal order. We give a brief illustrating example. Let f(n; ....

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P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Electronic J. Comb. 1 (1994) # R10.

.... ( Gamma1) k q k(5k Gamma3) 2 = 1 (q 2 ; q 5 ) 1 (q 3 ; q 5 ) 1 (6) 3 and e 1 = q; q) Gamma1 1 X k ( Gamma1) k q k(5k 1) 2 = 1 (q; q 5 ) 1 (q 4 ; q 5 ) 1 : 7) We remark that the proof of the crucial property (5) nowadays can be done automatically with the computer [8]. Also, note that the convertion of the series into the product representation is by Jacobi s triple product identity. See, e.g. 2, p. 21, eq. 2.2.10) We conclude this section by listing the first sequence entries explicitly. Namely, d 0 = 1; d 1 = 0; d 2 = 1; d 3 = 1; d 4 = 1 q 2 ; d ....

P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Electronic Journal of Combinatorics 1 (1994) R10, 1--9.

.... q q ) SUM[ 2 n] 1 q ) 1 q ) 2 2 n n 2 n 1 2 n 1 3 n (q (1 q q ) 1 q q q ) n 1 2 n SUM[ 1 n] 1 q ) 1 q ) The algorithm qHyper now finds the q hypergeometric solution of this recurrence (after replacing q n by x, and SUM[n k] by Y[qk x] In[10]: SUM[n k. Y[qk x] q(a. n b. xa qb; In[11] qHyper[ Y[x] Warning: irreducible factors of degree 1 in leading coefficient; some solutions may not be found 15 Warning: irreducible factors of degree 1 in trailing coefficient; some solutions may not be found 2 2 2 ....

....means that for one solution y n = Y (q n ) we have y n 1 y n = 1 q n 1 ) 1 Gamma q 2n 1 ) From this together with the initial values the right hand side evaluation of (31) is easily computed. Another way to treat the increase of recurrence orders in the q case was found by P. Paule [10]. His method of summing the even part , or variations of it, consists in rewriting the given sum by exploiting symmetries of the summand. After this preprocessing the q version of Zeilberger s algorithm delivers the recursion of minimal order. We give a brief illustrating example. Let f(n; k) ....

[Article contains additional citation context not shown here]

P. Paule, Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type, Electronic J. Comb. 1 (1994) # R10.

....(i) is provided by the q analogue of Petkovsek s algorithm Hyper; see Abramov, Paule and Petkovsek [1995] Nevertheless, creative symmetrizing in many instances solves all of these problems. A Mathematica q Analogue of Zeilberger s Algorithm 23 Creative symmetrizing has been introduced by Paule [1994] for certifying finite versions of Rogers Ramanujan type identities, for which the same problem as for (15) was observed. As found by Petkovsek, the method also can be applied in analogous q = 1 situations; see the book by Petkovsek, Wilf, and Zeilberger [1996] in which the term creative ....

....Checking equality at the initial values at n = 0 and n = 1 completes the proof of (16) Note that the proof certificates for both sides again are sufficiently nice, so that the verification of (12) can be easily done by a human. The qZeil proof of the companion identity to (16) can be found in Paule [1994]. We want to remark that identity (16) and its companion are the specializations a = 1 and a = q of X k a k q k 2 (q; q) k (q; q) n Gammak = X k ( Gamma1) k a 2k q k(5k Gamma1) 2 (aq; q) k Gamma1 (q; q) n Gammak (aq; q) n k (q; q) k Delta (1 Gamma aq 2k ) 17) a ....

Paule, P. [1994] Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type, Elec. J. Combinatorics, 1, R10.

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P. Paule. Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electron. J. Combin., 1, 1994. # R10.

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