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Horizontal and vertical generating functions and recursion relations have been investigated by Comtet for triangular double sequences. In this paper we investigate the horizontal and vertical log-concavity of triangular sequences assigned to polynomials which show up in combinatorics, number theory and physics. This includes Laguerre polynomials, the Pochhammer polynomials, the D'Arcais and Nekrasov–Okounkov polynomials. [ABSTRACT FROM AUTHOR]

We introduce a construction of a set of code sequences {Cn(m) : n ≥ 1, m ≥ 1} with memory order m and code length N(n). {Cn(m)} is a generalization of polar codes presented by Arıkan in [1], where the encoder mapping with length N(n) is obtained recursively from the encoder mappings with lengths N(n − 1) and N(n − m), and {Cn(m)} coincides with the original polar codes when m = 1. We show that {Cn(m)} achieves the symmetric capacity I(W) of an arbitrary binary-input, discrete-output memoryless channel W for any fixed m. We also obtain an upper bound on the probability of block-decoding error Pe of {Cn(m)} and show that Pe=O(2−Nβ) is achievable for β < 1/[1+m(ϕ − 1)], where ϕ ∈ (1, 2] is the largest real root of the polynomial F(m, ρ) = ρm − ρm − 1 − 1. The encoding and decoding complexities of {Cn(m)} decrease with increasing m, which proves the existence of new polar coding schemes that have lower complexity than Arıkan's construction. [ABSTRACT FROM AUTHOR]

For a signed cyclic graph G , we can construct a unique virtual link L by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F [ G ] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F [ G ] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally, we give a spanning subgraph expansion for F [ G ]. [ABSTRACT FROM AUTHOR]

We investigate the time evolution of Hagedorn wavepackets by non-Hermitian quadratic Hamiltonians. We state a direct connection between coherent states and Lagrangian frames. For the time evolution, a multivariate polynomial recursion is derived that describes the activation of lower lying excited states, a phenomenon unprecedented for Hermitian propagation. Finally we apply the propagation of excited states to the Davies–Swanson oscillator. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition is established for 1, cos(πr1), and cos(πr2) to be rationally independent, where r1, r2 are rational numbers. The elementary computational argument yields linear independenceover larger number fields as well. [ABSTRACT FROM AUTHOR]

The recurrence relation for the coefficients of higher order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a multiple composite function, is proved. Therefore, starting from this recurrence relation and by using the computer algebra program Mathematicac, some tables for complete higher order Bell polynomials and the relevant numbers are derived. [ABSTRACT FROM AUTHOR]

The Tamari lattices have been intensely studied since their introduction by Dov Tamari around 1960. However oddly enough, a formula for the number of maximal chains is still unknown. This is due largely to the fact that maximal chains in the n th Tamari lattice T n range in length from n − 1 to n 2 . In this note, we treat vertices in the lattice as Young diagrams and identify maximal chains as certain tableaux. For each i ≥ − 1 , we define C i ( n ) as the set of maximal chains in T n of length n + i . We give a recursion for # C i ( n ) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3 i + 3 . For example, the number of maximal chains of length n in T n is # C 0 ( n ) = n 3 . The formula has a combinatorial interpretation in terms of a special property of maximal chains. [ABSTRACT FROM AUTHOR]

We define and study the category of symmetric sl2-webs. This is a combinatorial description of the category of all finite-dimensional quantum sl2-modules. Explicitly, we show that (the additive closure of) the category of symmetric sl2-webs is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and colored Jones polynomials. [ABSTRACT FROM AUTHOR]

Let π ( G , x ) = ∑ i = 0 n b i x n − i be the permanental polynomial of a graph G , and π s ( G σ , x ) = ∑ i = 0 n d i x n − i the skew-permanental polynomial of an orientation graph G σ . In this paper, we investigate first the orientation graph G σ of a bipartite graph G satisfying | d n | = | b n | . Furthermore, we characterize the orientation of a bipartite graph with | d i | = | b i | for each i . Then we establish the recursion formulas for the skew-permanental polynomials of orientation graphs. In addition, we characterize the graphs whose skew-permanental polynomials can be expressed by the corresponding characteristic polynomials. [ABSTRACT FROM AUTHOR]

The Manakov hierarchy associated with amatrix spectral problem is proposed with the aid of Lenard recursion equations. By using the characteristic polynomial of Lax matrix for the Manakov hierarchy, we introduce a trigonal curveof arithmetic genus, from which we construct the related Baker–Akhiezer function, two algebraic functions carrying the data of the divisor and Dubrovin-type equations. Based on the theory of trigonal curves, the explicit theta function representations of the Baker–Akhiezer function, the two algebraic functions, and in particular, that of solutions for the entire Manakov hierarchy are obtained. [ABSTRACT FROM AUTHOR]

This paper concerns the distribution in the complex plane of the roots of a polynomial sequence { W n ( x ) } n ≥ 0 given by a recursion W n ( x ) = a W n − 1 ( x ) + ( b x + c ) W n − 2 ( x ) , with W 0 ( x ) = 1 and W 1 ( x ) = t ( x − r ) , where a > 0 , b > 0 , and c , t , r ∈ R . Our results include proof of the distinct-real-rootedness of every such polynomial W n ( x ) , derivation of the best bound for the zero-set { x | W n ( x ) = 0 for some n ≥ 1 } , and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory. [ABSTRACT FROM AUTHOR]

This paper analyzes and solves an integral and its indefinite Fourier transform of importance in multiple scattering problems of randomly distributed scatterers. The integrand contains a radiating spherical wave, and the two-dimensional domain of integration excludes a circular region of varying size. A solution of the integral in terms of radiating spherical waves is demonstrated. The method employs the Erdélyi operators, which leads to a recursion relation. This recursion relation is solved in terms of a finite sum of radiating spherical waves. The solution of the indefinite Fourier transform of the integral contains the indefinite Fourier transforms of the Legendre polynomials, which are solved by a closed formula. [ABSTRACT FROM AUTHOR]

We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q [ x ] , and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich. [ABSTRACT FROM AUTHOR]

By means of a new stability result, established for symmetric and multi-additive mappings, and using the concepts of stability couple and of stability chain, we prove, by a recursive procedure, the generalized stability of two of Fréchet's polynomial equations. We also give a new functional characterization of generalized polynomials and a new approach to solving the generalized stability of the monomial equation. [ABSTRACT FROM AUTHOR]

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a+b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of beta distributions. [ABSTRACT FROM AUTHOR]

We utilise a new approach via the so-called re-scaling method to derive a thorough theory for polynomial Riccati differential equations in the complex domain. [ABSTRACT FROM AUTHOR]

A packing function on a set Ω in Rn is a one-to-one correspondence between the set of lattice points in Ω and the set N0 of non-negative integers. It is proved that if r and s are relatively prime positive integers such that r divides s - 1, then there exist two distinct quadratic packing polynomials on the sector {(x, y) ∈ R2 : 0 ≤ y ≤ rx/s}. For the rational numbers 1/s, these are the unique quadratic packing polynomials. Moreover, quadratic quasi-polynomial packing functions are constructed for all rational sectors. [ABSTRACT FROM AUTHOR]

In this paper, we describe the recursion relations between two parameter HOMFLY and Kauffman polynomials of framed links. These relations correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings gn ⊃ gk× sln-k where gn is either SO2n+1, SO2n or sp2n is new. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study an integrable generalization of the associated Camassa–Holm equation. The generalized system is shown to be integrable in the sense of Lax pair and the bilinear Bäcklund transformations are presented through the Bell polynomial technique. Meanwhile, its infinite conservation laws are constructed, and conserved densities and fluxes are given in explicit recursion formulas. Furthermore, a Darboux transformation for the system is derived with the help of the gauge transformation between two Lax pairs. As an application, soliton and periodic wave solutions are given through the Darboux transformation. [Copyright &y& Elsevier]

A Euler{Seidel matrix is determined by an infinite sequence whose elements are given by recursion. The recurrence relations are investigated for numbers and polynomials such as hyperharmonics, Lucas numbers, and Euler and Genocchi polynomials. Linear recurring sequences in finite fields are employed, for instance, in coding theory and in several branches of electrical engineering. In this work, we define the period of a Euler{Seidel matrix over a field 픽p with p elements, where p is a prime number. We give some results for the matrix whose initial sequence is {sr(n)}n=0∞, where sr(n) = ..., n ≥ 0 and r is a fixed positive number. The numbers sr(n) play an important role in combinatorics and number theory. These numbers are known as 픽ranel numbers for r = 3. [ABSTRACT FROM AUTHOR]

Abstract: The objective of this paper is to prove that the initial Pola setting, with both inductive and coinductive data, is sound for polynomial size (PSIZE). Explicitly this means all programs written in Pola have their output size bounded by a polynomial in their input size. The paper describes the polarized categorical semantics for Pola and establishes the result by providing categorical models for various fragments of Pola which have explicit size bound information built into the maps. To obtain PSIZE soundness for Pola with just inductive data, the semantics in sized sets suffices. Sized sets are sets equipped with a “size map” which associates to each element a size. Size is usually just a natural number but, more generally, the size could be an element of a size rig. A polarized category consists of an opponent and a player category joined by a module. Sized sets can be used to create a polarized category by letting the opponent and module maps be bounded by polynomials while the player category consists of maps bounded by a constant. This gives a Pola setting with inductive data and immediately establishes the PSIZE soundness of the initial Pola setting. The main technical difficulty of the paper is to provide a semantics which correctly models coinductive data as well. For this “amortized” sets are introduced: these are set in which a higher-order size function is associated to each element, which given a size returns a size. This is amortized as one is concerned with the asymptotic behavior of these functions. While amortized sets have coinductive data they are not affine closed: a final step, using equivalence relations, is required to obtain a model which includes this aspect of Pola structure. While PSIZE by itself is a very weak bound it is a crucial step in establishing polynomial space (PSPACE) and polynomial time (PTIME) bounds for these settings. [Copyright &y& Elsevier]

Theon's ladder is an ancient method for easily approximating roots of a real number k. We extend previous work in this area by analysing a modification of this method for approximating roots of polynomials of the form x2 + cx + d. In particular, we will focus on establishing to which root the ladder will converge. We use techniques from real analysis such as monotone sequences and continuity of rational functions to prove the main results. [ABSTRACT FROM AUTHOR]

Real number block codes derived from the discrete Fourier transform (DFT) are corrected by coupling a very modified Berlekamp-Massey (BM) algorithm with a syndrome extension process. The modified BM algorithm determines recursively the locations of any large errors whose number is within the capability of the DFT code. It evolves a connection polynomial which is changed when a discrepancy is above a threshold which is adjusted during each iteration. The large error locations are repositioned to exact location indices to combat low-level noise effects. The syndromes are extended using the refined connection polynomial taps. Alternately, enhanced extension recursions based on Kalman syndrome extensions are developed and simulated. [ABSTRACT FROM AUTHOR]

Linear and zigzag polyacene series have been the subject of numerous studies because of their contrasting electronic and stability characteristics. The correlation of the properties of these series is examined in regard to their number of Dewar resonance structures (DS). Since resonance-theoretic methods require algorithms for determining the number of Dewar resonance structures (DS), recursion equations for calculating DS for these series are presented for the first time. Excellent correlations between DS and the absorption p-band, ionization energies, Hückel HOMO, Aihara's reduced HOMO-LUMO gap, topological resonance energy (TRE), aromatic stabilization energy (ASE), and the Klein and Randi6 innate degree of freedom are presented and rationalized. [ABSTRACT FROM AUTHOR]

Abstract: We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formula for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions. [Copyright &y& Elsevier]

Abstract: For a tridiagonal, singular matrix A we present a method for the computation of the polynomial such that holds, where is the Drazin inverse of A. The approach is based on the recursion of characteristic polynomials of leading principal submatrices of A. [Copyright &y& Elsevier]

We find an explicit expression of the Tutte polynomial of an n-fan. We also find a formula of the Tutte polynomial of an n-wheel in terms of the Tutte polynomial of n-fans. Finally, we give an alternative expression of the Tutte polynomial of an n-wheel and then prove the explicit formula for the Tutte polynomial of an n-wheel. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an algorithm to construct -point (for any integer ) binary approximating subdivision schemes has been developed using the Cox–de Boor recursion formula. Some properties like symmetry of basis functions and polynomial reproduction are also discussed. It can be observed that most of the existing binary approximating schemes (corner cutting scheme developed by Chaikin, and 3-point, 4-point, 5-point and 6-point schemes introduced by Siddiqi and Ahmad) are either special cases or can be reproduced by the proposed algorithm. [Copyright &y& Elsevier]

Golomb showed that fork, a fixed positive integer, one solution of the nested recursionis the Beatty sequence, whereis the positive root of the quadratic polynomial. We prove a natural generalization of this result: ifris any real root of some polynomial of degreed>1 with integer coefficients, then the Beatty functionis a solution of a ‘d-nested’ analogue of Golomb's original ‘2-nested’ recursion. [ABSTRACT FROM PUBLISHER]

Abstract: Integrability of polynomial differential system for any rational resonance ratio is studied. A new linear recursion formula for computation of generalized focus quantities and certain relation between two quantities from different definitions are given. As the applications, we discuss the generalized center conditions for two cubic Kolmogorov systems which are considered for the first time. [Copyright &y& Elsevier]

Based on solving the Lenard recursion equation and the zero-curvature equation, we derive the modified Sawada-Kotera (SK) hierarchy associated with a 3 × 3 matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix for the modified SK hierarchy, we introduce a trigonal curve Km-1 and present the corresponding Baker-Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the modified SK flows. With the aid of the property of the Baker-Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. Algebro-geometric solutions of the entire modified SK hierarchy are obtained by using asymptotic expansion of the meromorphic function and its Riemann theta function representation. As an application, we give two simple examples. [ABSTRACT FROM AUTHOR]

Abstract: The maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm. [Copyright &y& Elsevier]

We give an explicit solution of the ADHM recursion formula conjectured by Chuang, Diaconescu, and Pan. This solution is closely related to the formula for the Hodge polynomials of Higgs moduli spaces conjectured by Hausel and Rodriguez-Villegas. We also solve the twisted motivic ADHM recursion formula. As a byproduct, we obtain a conjectural formula for the motives of twisted Higgs moduli spaces, which generalizes the conjecture of Hausel and Rodriguez-Villegas. [ABSTRACT FROM AUTHOR]

Abstract: We show that the Poincaré polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the Eynard–Orantin type. The recursion uniquely determines the Poincaré polynomials from the initial data. Our key discovery is that the Poincaré polynomial is the Laplace transform of the number of Grothendieck’s dessins d’enfants. [Copyright &y& Elsevier]

Abstract: In this paper, we develop new numeric modified Adomian decomposition algorithms by using the Wazwaz–El-Sayed modified decomposition recursion scheme, and investigate their practicality and efficiency for several nonlinear examples. We show how we can conveniently generate higher-order numeric algorithms at will by this new approach, including, by using examples, 12th-order and 20th-order numeric algorithms. Furthermore, we show how we can achieve a much larger effective region of convergence using these new discrete solutions. We also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th-order numeric modified decomposition algorithm over the classic 4th-order Runge–Kutta algorithm by example. The efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by Duan for generation of the Adomian polynomials to high orders. [Copyright &y& Elsevier]

The bigraded Frobenius characteristic of the Garsia-Haiman module M is known [7, 10] to be given by the modified Macdonald polynomial $${\tilde{H}_{\mu}[X; q, t]}$$. It follows from this that, for $${\mu \vdash n}$$ the symmetric polynomial $${{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]}$$ is the bigraded Frobenius characteristic of the restriction of M from S to S. The theory of Macdonald polynomials gives explicit formulas for the coefficients c occurring in the expansion $${{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}$$. In particular, it follows from this formula that the bigraded Hilbert series F μ ( q, t) of M may be calculated from the recursion $${F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}$$. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that $${F\mu (q, t) \in \mathbf{N}[q, t]}$$. This difficulty arises from the fact that the c have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F( q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in []. [ABSTRACT FROM AUTHOR]

This paper presents two kinds of division polynomials for twisted Edwards curves. Their chief property is that they characterise the n-torsion points of a given twisted Edwards curve. We present recursions for the division polynomials, which differ in their flavour. We prove a uniqueness form for elements of the function field of an Edwards curve. We also present results concerning the coefficients of these polynomials, which may aid computation. [ABSTRACT FROM AUTHOR]

Abstract: For an arbitrary cocompact hyperbolic Coxeter group with a finite generator set and a complete growth function , we provide a recursion formula for the coefficients of the denominator polynomial . It allows us to determine recursively the Taylor coefficients and to study the arithmetic nature of the poles of the growth function in terms of its subgroups and exponent variety. We illustrate this in the case of compact right-angled hyperbolic -polytopes. Finally, we provide detailed insight into the case of Coxeter groups with at most 6 generators, acting cocompactly on hyperbolic 4-space, by considering the three combinatorially different families discovered and classified by Lannér, Kaplinskaya and Esselmann, respectively. [Copyright &y& Elsevier]

We give an algebraic proof of the formula on the norm of logarithmic primary of Virasoro algebra, which was proposed by Al. Zamolodchikov. This formula appears in the recursion formula for the norm of Gaiotto state, which guarantees the AGT relation for the four-dimensional SU(2) pure gauge theory. [ABSTRACT FROM AUTHOR]

We introduce a multiplicity Tutte polynomial M(x, y), with applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recursion and has positive coefficients. The characteristic polynomial and the Poincar´e polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x, y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1, y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x, 1) computes the volume and the number of integer points of the associated zonotope. [ABSTRACT FROM AUTHOR]

Abstract: We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof. [Copyright &y& Elsevier]

Conservative subtheories of $${{R}^{1}_{2}}$$ and $${{S}^{1}_{2}}$$ are presented. For $${{S}^{1}_{2}}$$, a slight tightening of Jeřábek's result (Math Logic Q 52(6):613-624, ) that $${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$$ is presented: It is shown that $${T^{0}_{2}}$$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this $${\forall\Sigma^{b}_{1}}$$-theory, we define a $${\forall\Sigma^{b}_{0}}$$-theory, $${T^{-1}_{2}}$$, for the $${\forall\Sigma^{b}_{0}}$$-consequences of $${S^{1}_{2}}$$. We show $${T^{-1}_{2}}$$ is weak by showing it cannot $${\Sigma^{b}_{0}}$$-define division by 3. We then consider what would be the analogous $${\forall\hat\Sigma^{b}_{1}}$$-conservative subtheory of $${R^{1}_{2}}$$ based on Pollett (Ann Pure Appl Logic 100:189-245, . It is shown that this theory, $${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$$, also cannot $${\Sigma^{b}_{0}}$$-define division by 3. On the other hand, we show that $${{S}^{0}_{2}+open_{\{||id||\}}}$$- COMP is a $${\forall\hat\Sigma^{b}_{1}}$$-conservative subtheory of $${R^{1}_{2}}$$. Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium' 98, 262-279, ) and show that $${\hat{C}^{0}_{2}}$$ is $${\forall\hat\Sigma^{b}_{1}}$$-conservative over a theory based on open-comprehension. [ABSTRACT FROM AUTHOR]

In this note we will introduce a class of search problems, called nested Polynomial Local Search (nPLS) problems, and show that definable NP search problems, i.e., $${\Sigma^{b}_{1}}$$-definable functions in $${T^{2}_{2}}$$ are characterized in terms of the nested PLS. [ABSTRACT FROM AUTHOR]

Abstract: We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations. [Copyright &y& Elsevier]

We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall--Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for the Hall--Littlewood functions, the theta polynomials of [[Buch, Kresch and Tamvakis, 2008]], and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w of the hyperoctahedral group and identify the set of reduced words for w with the set of standard k-tableaux on a skew shape λλ/μμ. [ABSTRACT FROM AUTHOR]

Abstract: Using the recursions satisfied by the polynomials which converge to the right-hand sides of the Rogers–Ramanujan type identities given by Sills (2003) and a determinant method presented in Ismail et al. (2000) , we obtain many new one-parameter generalizations of the Rogers–Ramanujan type identities, such as a generalization of the analytic versions of the first and second Göllnitz–Gordon partition identities, and generalizations of the first, second, and third Rogers–Selberg identities. [Copyright &y& Elsevier]

Abstract: Few scientific studies have discussed the accuracy of the Krawtchouk moments for the common case of p ≠0.5. In the paper, a novel symmetry and bi-recursive algorithm is proposed to accurately calculate the Krawtchouk moments for the case of p ∈ (0, 1). The numerical propagation error mechanism of direct recursively calculating the Krawtchouk moments is first analyzed. It reveals that the recursion coefficients and recurrence times of the three-term recurrence relations are the key factors of reducing the propagation error in the computation of the Krawtchouk moment of high order. Based on the analysis, the x − n plane is divided into four parts by x = n and x + n = N −1. We use the n-ascending recurrence formula to calculate the polynomials in the domain of N −1− n ⩾ x ⩾ n ⩾0 and apply the n-descending recurrence relations in the domain of 0⩽ N −1− n ⩽ x ⩽ n. Thus the maximum recursion times are limited to N/2. Finally, with the help of the diagonal symmetry property on x = n, the Krawtchouk polynomial values of high precision in the whole x − n coordinates are obtained. The algorithm ensures that the maximum recursive numerical errors are within an acceptable range. An experiment on a large image of 400×400pixels is designed to demonstrate the performance of the proposed algorithm against the classical method. [Copyright &y& Elsevier]

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order. A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved. The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory. [ABSTRACT FROM AUTHOR]

Abstract: K. Dohmen, A. Pönitz and P. Tittmann [K. Dohmen, A. Pönitz, P. Tittmann, A new two-variable generalization of the chromatic polynomial, Discrete Mathematics and Theoretical Computer Science 6 (2003), 69–90], introduced a bivariate generalization of the chromatic polynomial which subsumes also the independent set polynomial of I. Gutman and F. Harary [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Mathematicae 24 (1983), 97–106] and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little [F.M. Dong, M.D. Hendy, K.L. Teo, and C.H.C. Little, The vertex-cover polynomial of a graph, Discrete Mathematics 250 (2002), 71–78]. We first show that has a recursive definition with respect to three kinds of edge eliminations: edge deletion, edge contraction, and edge extraction, i.e. deletion of an edge together with its endpoints. Like in the case of deletion and contraction only [J.G. Oxley and D.J.A. Welsh, The Tutte polynomial and percolation, in: J.A. Bundy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, London, 1979, pp. 329–339] it turns out that there is a most general, or as they call it, a universal polynomial satisfying such recurrence relations with respect to the three kinds of edge eliminations, which we call . We show that the new polynomial simultaneously generalizes, , as well as the Tutte polynomial and the matching polynomial, We also give an explicit definition of using a subset expansion formula. We also show that can be viewed as a partition function, using counting of weighted graph homomorphisms. Furthermore, we expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky [T. Zaslavsky, Strong Tutte functions of matroids and graphs, Trans. Amer. Math. Soc. 334 (1992), 317–347] and by B. Bollobás and O. Riordan [B. Bollobás, O. Riordan, A Tutte polynomial for coloured graphs, Combinatorics, Probability and Computing 8 (1999), 45–94]. The edge-labeled polynomial also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. [R.C. Read, E.G. Whitehead Jr., Chromatic polynomials of homeomorphism classes of graphs, Discrete Mathematics 204 (1999), 337–356]. Finally, we discuss the complexity of computing . [Copyright &y& Elsevier]