Two new multivariable generating relations

A multidimensional extension of Bailey's transform is utilised to deduce two new generating relations of quite a general character. These expressions are then specialised to give more practical formulae in terms of Karlsson's generalised Kampé de Fériet functions which embody very many generating relations. A number of interesting special cases are given in an appendix which includes results involving Lauricella polynomials, generalised hypergeometric polynomials and the polynomials of Meixner, Charlier and Laguerre.     

Two variable Lagrange expansion and new families of mixed generating functions

We show that a new method based on the combination of the two variable Lagrange expansion and of operational techniques can be exploited to derive new families of mixed generating functions for many variable many index special functions.

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Sunto In questo lavoro si fa uso di un nuovo metodo, basato sulla combinazione della espansione bidimensionale di Lagrange e di tecniche operazionali, per ottenere nuove famiglie di funzioni generatrici miste per funzioni speciali a più indici e a più variabili.

     

Extension of certain types of generating relations

This paper gives a new expansion formula which essentially involves a double sum. As a consequence of our main result, eq. (6), various types of generating relations are seen to emerge and relevance with some known results is pointed out briefly. The usefulness of our main result is also indicated by considering its application to a probabilistic method for a lattice path enumeration problem.     

Lie-theoretic generating relations of Hermite 2D polynomials

In this paper, we derive some generating relations involving Hermite 2D polynomials (H2DP) H_m,n(U;x,y), of two variables with an arbitrary 2D matrix U as parameter using Lie-theoretic approach. Certain (known or new) generating relations for the polynomials related to H2DP are also obtained as special cases.     

Certain bilateral generating relations for generalized hypergeometric functions

Recently, we introduced a class of generalized hypergeometric functionsI n:(b _q)/α:(a _p) (x, w) by using a difference operator Δ _x,w , where . In this paper an attempt has been made to obtain some bilateral generating relations associated withI _n ^ga (x, w). Each result is followed by its applications to the classical orthogonal polynomials.     

On the Galilean group for balance relations generating kinetic schemes

The conditions are considered for the invariance under the Galilean transformation group of balance relations used in kinetically consistent difference schemes.     

On generating regular elements in the semigroup of binary relations

We prove the semigroup generated by four binary relations contains all regular binary relations.     

Radon’s construction and matrix relations generating syzygies

Monatshefte für Mathematik - Let $$varPi _n$$ be the set of bivariate polynomials of degree not greater than n. A $$varPi _n$$-correct set of nodes is a set such that the Lagrange...     

On a set of generating relations of the full transformation semigroups

A set of generating relations of the full transformation semigroup Σ_n consisting of all mappings of the set $\text{Ω = {1, …, n}}$ into itself with respect to a natural system of generators {γ_jⁱ, σ_jⁱ} (defined in Section 1) is given. The set of relations (1)～(15) given in Section 1 turns out to be a set of generating relations.     

A confinal family of equivalence relations and Borel ideals generating them

An increasing θ₁-sequence of Borel equivalence relations on a Polish space that is cofinal (in the sense of Borel reducibility) in the family of all Borel equivalence relations is defined as a development of Rosendal’s construction. It is proved that equivalence relations from this sequence are generated by explicitly defined Borel ideals.     

A new algorithm for generating derangements

A new algorithm for generating derangements based on a well known permutation generation method is presented and analysed. The algorithm is shown to be superior in storage and time requirements to the existing method.This work was supported by the Natural Sciences and Engineering Research Council of Canada under Grant NSERC-A3336.     

Grey multivariable convolution model with new information priority accumulation

The GM(1,N) model with convolution integral appeals considerable interest in recent researches due to its effectiveness in multivariate time series forecasting. However, the failure of incorporation new information priority principle will cause large errors. To improve the simulation and prediction accuracy, GMC(1,N) model with new information priority accumulation is put forward. A parameter is added to adjust the weight of data. By giving a large weight to the new information, the accuracy of the prediction is improved in theoretical. The priority of new GMC(1,N) model is verified through some cases.     

A new multivariable grey prediction model with structure compatibility

A new multivariable grey prediction model was proposed by adding a dependent variable lag term, a linear correction term and a random disturbance term to the traditional GM(1,N) model. It was theoretically proved that the new model can be completely compatible with the mainstream single variable and multivariable grey prediction models by adjusting and changing the model's parameters. To test the performance of the new model, three case studies were performed. The simulation and prediction results of the new model were compared with those of other grey prediction models. Results showed that the new model had evidently superior performance to other grey models, which confirms that the structure design of the new model is more reasonable than those of the other existing grey prediction models.     

An optimal algorithm for generating equivalence relations on a linear array of processors

We describe a cost-optimal parallel algorithm for enumerating all partitions (equivalence relations) of the set {1, ...,n}, in lexicographic order. The algorithm is designed to be executed on a linear array of processors. It usesn processors, each having constant size memory and each being responsible for producing one element of a given set partition. Set partitions are generated with constant delay leading to anO(B _n) time solution, whereB _n is the total number of set partitions. The same method can be generalized to enumerate some other combinatorial objects such as variations. The algorithm can be made adaptive, i.e. to run on any prespecified number of processors. To illustrate the model of parallel computation, a simple case of enumerating subsets of the set {1, ...,n}, having at mostm (n) elements is also described.The research is partialy supported by NSERC operating grant OGPIN 007.     

A note on certain bilinear and bilateral generating relations for the laguerre polynomial

The author presents here a systematic analysis of certain bilinear and bilateral generating relations associated with the generalized Laguerre polynomials {L_n ^(a) (x)} defined by $$(1 - t)^{ - 1 - \alpha } \exp \left( { - \frac{{xt}}{{1 - t}}} \right) = \sum\limits_{n = 0}^\infty {L_n ^{(\alpha )} (x) t^n .} $$ It is observed that the four formulas proved recently by Jain [7] are only specialized or limiting forms of the results of Erdélyi,[4] Meixner [8] and the author [12–14].     

Generalized polynomials and new families of generating functions

We show that the use of generalized polynomials, i.e. polynomials defined as discrete convolution of Hermite, Laguerre… polynomials can be exploited to explore new families of generating functions.

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Sunto Si dimostra che l'uso di polinomi generalizzati definiti come convoluzioni discrete di polinomi di Hermite, Laguerre etc. possono essere utilizzati per studiare nuove famiglie di funzioni generatrici.

     

Two relations between oblique and Λ-orthogonal projectors

Let $\text{L}$ and $\text{M}$ be any complementary subspaces. In this article, two relations established by T.N.E. Greville between the projector P_{$\text{L}$|$\text{M}$} on $\text{L}$ along $\text{M}$ and the orthogonal projectors on $\text{L}$ and $\text{M}$ are generalized by admitting any Λ-orthogonal projectors, with Λ being a positive definite matrix. Also, two representations of Λ are found for which, given $\text{L}$ and $\text{M}$, Λ-orthogonal projectors on $\text{L}$ become identical with P_{$\text{L}$|$\text{M}$}.