Strong factorization property of Macdonald polynomials and higher-order Macdonald’s positivity conjecture

We prove a strong factorization property of interpolation Macdonald polynomials when q tends to 1. As a consequence, we show that Macdonald polynomials have a strong factorization property when q tends to 1, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate q, t-Kostka numbers and we show that they are polynomials in q, t with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate q, t-Kostka numbers are in fact polynomials in q, t with nonnegative integer coefficients, which generalizes the celebrated Macdonald’s positivity conjecture.     

Zonal spherical functions on CROSS’s and special functions

We find formulas for the eigenvalues of the Laplacian and the zonal spherical functions on all simply-connected CROSS??s by a simple method, using the trigonometric formulas of spherical geometry, Hopf fiber bundles, and the results on the spectra of the Laplacian on the total space and on the base of a Riemannian submersion with totally geodesic fibers. We find direct relations of the so-obtained zonal spherical functions to the special functions: hypergeometric finite Gauss series, Jacobi polynomials, and orthogonal polynomials including the ultraspherical Gegenbauer polynomials whose particular cases are given by the Legendre polynomials and the Chebyshev polynomials of the first and second kinds. We point out the relations to the corresponding results by Helgason and Berger with coauthors and give brief information about the method of calculating the spectra of the Laplacian on compact simply-connected irreducible Riemannian spaces and the spectra of the Laplacian on the CROSS??s obtained therefrom.     

Skew Pieri rules for Hall–Littlewood functions

We produce skew Pieri rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (J. Comb. Theory Ser. A 118(1):277–290, 2011). The first two were conjectured by the first author (Konvalinka in J. Algebraic Comb. 35(4):519–545, 2012). The key ingredients in the proofs are a q-binomial identity for skew partitions and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and the antipode.     

Rademacher-type formulas for restricted partition and overpartition functions

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.     

A Nekrasov–Okounkov formula for Macdonald polynomials

We construct the irreducible unipotent modules of the finite general linear groups from actions on tableaux. Our approach is analogous to that of James (Bull Lond Math Soc 8:229–232, 1976) for the symmetric groups, answering an open question as to whether such a construction exists. We show that our modules are isomorphic to those previously constructed by James (Representations of general linear groups, London Mathematical Society Lecture Note Series, vol. 94. Cambridge University Press, Cambridge, 1984. doi: 10.1017/CBO9780511661921) , although the two presentations are quite different. Key to our construction are the generalized Gelfand–Graev representations of Kawanaka (Generalized Gel’fand-Graev representations and Ennola duality. In: Algebraic groups and related topics (Kyoto/Nagoya, 1983), advanced studies in pure math., vol. 6, pp. 175–206. North-Holland, Amsterdam 1985).     

A Littlewood–Richardson rule for Macdonald polynomials

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric Macdonald polynomials are a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general Lie type.     

Haglund’s conjecture on 3-column Macdonald polynomials

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund (Proc Natl Acad Sci USA 101(46):16127–16131, 2004). The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam’s (Eur J Combin 29(1):343–359, 2008) algebra of ribbon Schur operators. Combining this result with the expression of Haglund et al. (J Am Math Soc 18(3):735–761, 2005) for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.     

Kibble–Slepian formula and generating functions for 2D polynomials

We prove a generalization of the Kibble–Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [16]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.     

Recurrence formulas for Macdonald polynomials of type <Emphasis Type=Italic>A</Emphasis>

We consider products of two Macdonald polynomials of type A, indexed by dominant weights which are respectively a multiple of the first fundamental weight and a weight having zero component on the kth fundamental weight. We give the explicit decomposition of any Macdonald polynomial of type A in terms of this basis.     

Narayana polynomials and Hall–Littlewood symmetric functions

We show that Narayana polynomials are a specialization of row Hall–Littlewood symmetric functions. Using λ-ring calculus, we generalize to Narayana polynomials the formulas of Koshy and Jonah for Catalan numbers.     

Modifications of Newton’s method for even-grade palindromic polynomials and other twined polynomials

The paper describes some modifications of Newton??s method for refining the zeros of even-grade f(x)-twined (f(x)-egt) polynomials, defined as polynomials whose roots appear in pairs {x _i ,f(x _i )}. Particular attention is given to even-grade palindromic (egp) polynomials. The algorithms are derived from certain symmetric division processes for computing a symmetric quotient and a symmetric remainder of two given f(x)-egt polynomials. Numerical results indicate that the presented algorithms can be more accurate than other methods which do not take into consideration the symmetry of the coefficients.     

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over and a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of . This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula. Received: April 2005 Revision: June 2006 Accepted: October 2006     

Summation formulas for the products of the Frobenius–Euler polynomials

We present here a further investigation for the classical Frobenius–Euler polynomials. By making use of the generating function methods and summation transform techniques, we establish some summation formulas for the products of an arbitrary number of the classical Frobenius–Euler polynomials. The results presented here are generalizations of the corresponding known formulas for the classical Bernoulli polynomials and the classical Euler polynomials.     

Some extensions for Ramanujan’s circular summation formulas and applications

The Ramanujan Journal - In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we...