## A $$q$$-rious positivity.(English)Zbl 1234.11023

The $$q$$-factorial of an integer $$n\geq 1$$ is defined by $[n]!=\prod_{i=1}^n\frac{1-q^i}{1-q}.$ For $${\mathbf a}=(a_1,\dots,a_r)$$ and $${\mathbf b}=(b_1,\dots,b_s)$$ tuples of positive integers satisfying $\sum_{i=1}^r\lfloor a_i x \rfloor-\sum_{j=1}^s\lfloor b_j x \rfloor\geq 0,\forall x\geq 0$ the authors conjecture that the polynomial $D({\mathbf a},{\mathbf b};q):=\frac{[a_1]!\dots[a_r]! }{[b_1]!\dots[b_s]!}$ is positive. The conjecture is true for the $$q$$-super Catalan numbers $A_{n,m}(q)=D((2n,2m),(n,n+m,m);q),n,m\geq 0$ and for $B_{n,m}(q):=D((2n,m),(n,2m,n-m);q),n\geq m\geq 0.$

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 05A10 Factorials, binomial coefficients, combinatorial functions 11B83 Special sequences and polynomials 11C08 Polynomials in number theory 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$

### Keywords:

$$q$$-binomial coefficients; polynomials; positivity
Full Text:

### References:

 [1] Andrews G.E.: On a conjecture of Peter Borwein. J. Symb. Comput. 20, 487–501 (1995) · Zbl 0849.68062 [2] Berkovich A., Warnaar S.O.: Positivity preserving transformations for q-binomial identities. Trans. Am. Math. Soc. 357, 2291–2351 (2005) · Zbl 1061.33015 [3] Bober J.W.: Factorial ratios, hypergeometric series, and a family of step functions. J. Lond. Math. Soc. (2) 79, 422–444 (2009) · Zbl 1195.11025 [4] Bressoud D.M.: The Borwein conjecture and partitions with prescribed hook differences. Electron. J. Comb. 3, 14 (1996) #R4 · Zbl 0856.05007 [5] Catalan E.: Nouvelles Annales de Mathématiques. J. Candidats École Polytech. Normale (2) 13, 207 (1874) [6] Delaygue, E.: Critère pour l’intégralité des coefficients de Taylor des applications miroir. J. Reine Angew. Math. (in press) arXiv: 0912.3776 [math.NT] (2009) [7] Gasper G., Rahman M.: Basic Hypergeometric Series. Encyclopedia Math. Appl., vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004) · Zbl 1129.33005 [8] Gessel I.: Super ballot numbers. J. Symb. Comput. 14, 179–194 (1992) · Zbl 0754.05002 [9] Guo V.J.W., Jouhet F., Zeng J.: Factors of alternating sums of products of binomial and q-binomial coefficients. Acta Arith. 127, 17–31 (2007) · Zbl 1116.05004 [10] Landau, E.: Sur les conditions de divisibilité d’un produit de factorielles par un autre. In: Collected Works, vol. I, 116 pp. Thales-Verlag, Essen (1985) [11] Picon, P.A.: Sur un théorème de Landau. Sém. Lotharingien Comb. 4 (1984), #B08k (formely: Publ. IRMA Strasbourg (1984), no. 229/S-08, pp. 75–78) [12] Pólya, G., Szego, G.: Problems and Theorems in Analysis, vol. II, Grundlehren Math. Wiss., vol. 216. Springer, Berlin (1976) [13] Proctor R.A.: Solution of two difficult combinatorial problems with linear algebra. Am. Math. Mon. 89, 721–734 (1982) · Zbl 0509.05007 [14] von Szily K.: Über die Quadratsummen der Binomialcoefficienten. Ungar. Ber. 12, 84–91 (1894) · JFM 25.0405.01 [15] Warnaar, S.O.: The generalized Borwein conjecture. I. The Burge transform. In: Berndt B.C., Ono K. (eds.) q-Series with Applications to Combinatorics, Number Theory, and Physics. Contemp. Math. vol. 291, pp. 243–267. Amer. Math. Soc., Providence (2001) · Zbl 0994.05009 [16] Warnaar S.O.: The generalized Borwein conjecture. II. Refined q-trinomial coefficients. Discrete Math. 272, 215–258 (2003) · Zbl 1030.05004 [17] Zeilberger D.: Kathy O’Hara’s constructive proof of the unimodality of the Gaussian polynomials. Am. Math. Mon. 96, 590–602 (1989) · Zbl 0726.05005 [18] Zudilin W.: On the irrationality measure for a q-analogue of {$$\zeta$$}(2). Russ. Acad. Sci. Sb. Math. 193, 1151–1172 (2002) · Zbl 1044.11067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.