G. E. Andrews. A polynomial identity which implies the Rogers-Ramanujan identities. Scripta Mathematica, 28:297-305, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the Rogers Ramanujan identities, 3.4) and Theorem 3.1, which are inverses of each other. Polynomials that Schur considered in his work on the RogersRamanujan continued fraction naturally appear. Moreover one is led to the finite forms of the RogersRamanujan identities due to Andrews [2]. Consider the integral I m (t; q) q; q) 1 2 Z 0 Hm (cos jq) te i ; te Gammai ; e 2i ; e Gamma2i ; q) 1 d : 3.1) 4 Clearly I 0 (t; q) I(t; q) As in the proof of (1.1) using q Hermite orthogonality we find that I m (t; q) is given by I m (t; q) 1 X l=0 ( Gammat) ....

....2 c # q : 3.6) These polynomials were considered by Schur [22] 8] 3] as numerators and denominators of the RogersRamanujan continued fraction. The left side of (3.5) is the generating function for partitions with difference at least two whose smallest part is at least m 1. Andrews [2] gave a polynomial generalization of the Rogers Ramanujan identities by showing that am (q) X j q j 2 j m Gamma j Gamma 2 j # q ; b m (q) X j q j 2 m Gamma j Gamma 1 j # q : 3.7) They have the following combinatorial interpretations: am (q) b m (q) is the ....

G. E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math. 28 (1970), 297--305.

....as a proof by qWZ certification, based on the method of creative telescoping. Insection3wepresentanew veryshort qWZ certification proof of (2) that was missed by the computers Ekhad and Tre [12] In section 4 we provide the qWZ certificate of another finite Rogers Ramanujan version due to Andrews [2] and which originates in the work of Schur [21] who independently discovered (1) As briefly indicated in section 4, this finite version plays an outstanding role for certain applications in physics. Besides the versions given in Andrews [8] all known finite Rogers Ramanujan type identities ....

..... Without introducing the extra symmetry factor 1 q k , the left hand side of (48) in Paule [17] a finite version of a Rogers Selberg identity, has minimal recursion order 7 ( instead of 3, with respect to qWZ certification. 4 The Andrews Schur Identity As observed by Andrews [2], the following po lynomial identity for a = 0 and a = 1 immediately implies (1) by taking n ##and using Jacobi s triple product identity (see the next section) the electronic journal of combinatorics 1 (1994) # R10 5 n # k=0 q k 2 ak # 2n k a k # = 13) # # k= # q 10k ....

[Article contains additional citation context not shown here]

G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan Identities, Scripta. Math., 28 (1970), 297--305.

....by qWZ certification, based on the method of creative telescoping. In section 3 we present a new very short qWZ certification proof of (2) that was missed by the computers Ekhad and Tre [12] In section 4 we provide the qWZ certificate of another finite Rogers Ramanujan version due to Andrews [2] and which originates in the work of Schur [21] who independently discovered (1) As briefly indicated in section 4, this finite version plays an outstanding role for certain applications in physics. Besides the versions given in Andrews [8] all known finite Rogers Ramanujan type identities ....

....ffl Without introducing the extra symmetry factor 1 q k , the left hand side of (48) in Paule [17] a finite version of a Rogers Selberg identity, has minimal recursion order 7 ( instead of 3, with respect to qWZ certification. 4 The Andrews Schur Identity As observed by Andrews [2], the following polynomial identity for a = 0 and a = 1 immediately implies (1) by taking n 1 and using Jacobi s triple product identity (see the next section) the electronic journal of combinatorics 1 (1994) # R10 5 n X k=0 q k 2 ak 2n Gamma k a k = 13) 1 X k= Gamma1 q ....

[Article contains additional citation context not shown here]

G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan Identities, Scripta. Math., 28 (1970), 297--305.

....the right side of (13) is, indeed, a q analog of ( Gamma1) F k , it mixes the variables m and k in such a way that ( Gamma1) is the only trace of m left when q = 1. ii) In fact, Proposition 1 is a natural generalization of the polynomial versions of the RogersRamanujan identities given in [1]. A computer proof can be found in [8, Theorem 2] Proposition 1 itself is a polynomial version of the Garrett, Ismail and Stanton result; namely, sending k to infinity in (13) immediately results in (8) Now we are ready for the proof of Theorem 3. 9 2.2 Proof of Theorem 3 Proof of Theorem 3. ....

G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math. 28 (1970), 297--305. 12

....1 ; 1.1) where a = 0; 1 and the q shifted factorials (z; q) t are de ned as usual as (z; q) t = z) t = Q t 1 j=0 (1 zq j ) if t 2 Z 0 ; 1; if t = 0: 1.2) It is well known that these identities have polynomial analogs. In particular, building on the work of Schur and MacMahon, Andrews [1] has shown that for L 2 Z 0 X t 0 q t 2 L t t q = e L (q) 1.3) and X t 0 q t 2 t L t 1 t q = dL (q) 1.4) where e L (q) 1 X j=1 ( q j(10j 1) L b L 2 c 5j q q (2j 1) 5j 2) L b L 4 2 c 5j q ) 1.5) and dL (q) 1 X j=1 ( q ....

G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math., 28 (1970), 297-305.

....right side of (13) is, indeed, a q analog of ( Gamma1) m F k , it mixes the variables m and k in such a way that ( Gamma1) m is the only trace of m left when q = 1. ii) In fact, Proposition 1 is a natural generalization of the polynomial versions of the RogersRamanujan identities given in [1]. A computer proof can be found in [8, Theorem 2] Proposition 1 itself is a polynomial version of the Garrett, Ismail and Stanton result; namely, sending k to infinity in (13) immediately results in (8) Now we are ready for the proof of Theorem 3. 2.2 Proof of Theorem 3 Proof of Theorem 3. ....

G.E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math. 28 (1970), 297--305.

No context found.

G. E. Andrews. A polynomial identity which implies the Rogers-Ramanujan identities. Scripta Mathematica, 28:297-305, 1970.

No context found.

G. E. Andrews. A polynomial identity which implies the Rogers-Ramanujan identities. Scripta Mathematica, 28:297--305, 1970.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC