𝑞-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra

New Identities for Hall-Littlewood Polynomials and Applications

“…However, as remarked by Stembridge [13, p. 475], in these other cases there arise complications which render doubtful the existence of expansions as explicit as those of (3) and (4). We noticed that these complications arise if one wants to keep exactly the same coefficients c λ (q) and d λ (q) as in (5) and (6) for the sums over bounded partitions.…”

𝑞-Hermite polynomials, biorthogonal rational functions, and 𝑞-beta integrals

“…The biorthogonal rational functions are p"(sinh£; ti,h,h, U) (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) (q-n,-ht2qn-2,-ht3lq,-hmq \ They satisfy the orthogonality relation (1.21) /OO <Pm{x; h,t2,h, U)<pn{x; t2,t{,t3, t4)w{x)dy/{x) = g"dm>n ,…”

“…To identify the functions involved we expand the right side of (3.1) using (3.2) This gives the summation formula (3)(4)(5)(6)(7)(8)(9) Y…”

“…In the first example we choose a{z) in Theorem 7.2 as (7)(8)(9) g(z):=l[,z + c__L_ , for real c and b > 0, so that a in (6.1) is <7i(z), (7.10) ax{z) = -B{{bz + c-{bz + c)~x)/2)/D{{bz + c-{bz + c)~x)/2).…”

“…Rogers [8], [44], [45], [46] introduced a very interesting system of polynomials and used it to prove the Rogers-Ramanujan identities. The polynomial system is {C"(x; f$ \q)} (1.6) C(cos0, fi\g) = J2 iM^±ei(n-2k)0_ £f0 \Q)k{Q)n-k…”