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1
Content available Lucas-type associated polynomials
Guettai Ghania, Laissaoui Diffalah, Rahmani Mourad
Mathematica Applicanda
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2023
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Vol. 51, no. 2
291--304
EN
In this paper, we define a new type of Lucas polynomials known as Lucas-type associated polynomials and investigate their fundamental properties and identities. An interesting formula for Lucas-type associated polynomials can be derived using Leibniz's rule for derivatives, defined by Rodrigue's Lucas-type formula. Additionally, we establish an integral connection between Lucas-type associated polynomials and associated Fibonacci polynomials.
PL
W tym artykule definiujemy nowy rodzaj wielomianów Lucasa, znanych jako wielomiany powiązane typu Lucasa, i badamy ich podstawowe właściwości i tożsamości. Ciekawy wzór na wielomiany powiązane typu Lucasa można uzyskać, używając reguły Leibniza dla pochodnych, zdefiniowanej przez wzór Rodrigue’a dla wielomianów powiązanych typu Lucasa. Dodatkowo ustanawiamy związek całkowy między wielomianami powiązanymi typu Lucasa a wielomianami Fibonacciego
2
Content available Optimal Strategies for Computation of Degree degree ℓⁿ Isogenies for SIDH
Wroński Michał, Chojnacki Andrzej
International Journal of Electronics and Telecommunications
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2020
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Vol. 66, No. 3
465--472
EN
This article presents methods and algorithms for the computation of isogenies of degree ℓⁿ. Some of these methods are obtained using recurrence equations and generating functions. A standard multiplication based algorithm for computation of isogeny of degree ℓⁿ has time complexity equal to O(n²M (n log n)), where M (N) denotes the cost of integers of size N multiplication. The memory complexity of this algorithm is equal to O (n log (n log (n))). In this article are presented algorithms for: - determination of optimal strategy for computation of degree ℓⁿ isogeny, - determination of cost of optimal strategy of computation of ℓⁿ isogeny using solutions of recurrence equations, - determination of cost of optimal strategy of computation of ℓⁿ isogeny using recurrence equations, where optimality in this context means that, for the given parameters, no other strategy exists that requires fewer operations for computation of isogeny. Also this article presents a method using generating functions for obtaining the solutions of sequences (սₘ) and (cₘ) where cₘ denotes the cost of computations of isogeny of degree ℓᵘᵐum for given costs p, q of ℓ-isogeny computation and ℓ-isogeny evaluation. These solutions are also used in the construction of the algorithms presented in this article.
3
Content available remote On a new generalization of Jacobsthal quaternions and several identities involving these numbers
Bród Dorota, Szynal-Liana Anetta
Commentationes Mathematicae
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2019
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Vol. 59, No. 1/2
33--45
EN
In this paper we generalize Jacobsthal quaternions to (s, p) -Jacobsthal quaternions. We give some of their properties, among others the Binet formula, the generating function and the matrix representation of these quaternions. We will show how a graph interpretation can be used in proving some identities for quaternions.
4
Content available remote Unified (p, q)-Bernoulli-Hermite polynomials
Pathan M. A.
Fasciculi Mathematici
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2018
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Nr 61
125--141
EN
The Concepts of p-Bernoulli numbers Bn,p and p-Bernoulli polynomials Bn,p(x) are generalized to (p,q)-Bernoulli numbers Bn p q and (p,q)-Bernoulli polynomials Bn p q(x), respec- tively. Some properties, generating functions and Laplace hy- pergeometric integral representations of (p, q)-Bernoulli numbers Bn,p,q and (p,q)-Bernoulli polynomials Bn,p,q(x), are established. Unified (p,q)-Bernoulli-Hermite polynomials are defined by a generating function which aid in proving the generalizations of the results of Khan et al [8], Kargin and Rahmani [7], Dattoli [4] and Pathan [9]. Some explicit summation formulas and some relationships between Appell’s function F1, Gauss hypergeomtric function, Hurwitz zeta function and Euler’s polynomials are also given.
5
Content available remote Generalized Stern polynomials and hyperbinary representations
Dilcher K., Ericksen L.
Bulletin of the Polish Academy of Sciences. Mathematics
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2017
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Vol. 65, no. 1
11--28
EN
We use two different but related types of generalized Stern polynomials, recently introduced by the authors, to give complete characterizations of all hyperbinary expansions of a given positive integer.We also derive explicit formulas for these generalized Stern polynomials and use them to establish further characterizations of hyperbinary expansions, using binomial coeffcients. We then introduce a 2-parameter analogue of the two types of polynomials, which leads to more explicit versions of earlier results. Finally, we explore further generalizations of the polynomials studied in this paper.
6
Content available remote Generalized elliptic-type integrals and generating functions
Chaurasia V. B. L., Singh Y.
Demonstratio Mathematica
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2014
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Vol. 47, nr 2
310--323
EN
On account of analytical importance or application in certain problems in radiation physics and nuclear technology, several interesting families of elliptic-type integrals were recently studied by many authors. The aim and objective of present paper is to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. In addition, to deriving known and various new elliptic-type integrals and their generalizations, these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.
7
Content available remote Dyck Paths with Parity Restrictions for the Final Runs to the Origin: a Study of the Height
Prodinger H.
Fundamenta Informaticae
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2012
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Vol. 117, nr 1-4
279-285
EN
Stanley and Callan considered Dyck paths where the lengths of the run to the origin is always odd resp. the last one even, and the other ones odd. These subclasses are also enumerated by (shifted) Catalan numbers. We study the (average) height of these objects, assuming all such Dyck paths of length 2n to be equally likely, and find that it behaves like ~ √πn, as in the unrestricted case. This classic result for unrestricted Dyck paths is from de Bruijn, Knuth and Rice [2], and to this day, there are no simpler proofs for this, although more general results have been obtained by Flajolet and Odlyzko [4].
8
Content available remote A note on generating functions and summation formulae for Meixner polynomials of several variables
Khan M. A., Akhlaq M.
Demonstratio Mathematica
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2012
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Vol. 45, nr 1
51-66
EN
The present paper deals with certain generating functions and various elegant summation formulae for Meixner polynomials of several variables.
9
Content available remote Ultraspherical type generating functions for orthogonal polynomials
Demni N.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 2
281--296
EN
We characterize, under some technical assumptions and up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials. Our method is based on differential equations and the obtained measures are particular beta distributions. We actually recover the free Meixner family of probability distributions so that our method gives a new approach to the characterization of free Meixner distributions.
10
Content available remote Binary Search Trees, Recurrent Properties and Wave Equations
Fernández-Camacho M-I., Sánchez-Couso J-R.
Fundamenta Informaticae
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2007
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Vol. 81, nr 4
409-439
EN
We give a generic framework to analyze the average-case running time for computing the so called recurrent properties for pairs of binary search trees. Recurrent properties are algorithms that operate on pairs of trees testing some characteristic on nodes by performing a preorder traversal on both trees. Analysis of recurrent properties using the probability model associated with randomly grown binary search trees leads to wave equations. We use a "normalized" integral equation as a pattern to model a specific wave equation and investigate the asymptotic behavior of its solution. This methodology is applied to some particular cases of recurrent properties like testing equality, detecting direct occurrences and clashes or pattern matching.
11
Content available remote Asymptotic Properties of the Factors of Words Over a Finite Alphabet
Tomescu I.
Fundamenta Informaticae
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2005
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Vol. 64, nr 1-4
463-470
EN
Let A be an alphabet of cardinality m, kn be a sequence of positive integers and w e A* (|w| = kkn). In this paper it is shown that if lim sup n→∞knn <1/lnm, then almost all words of length n over A contain the factor w, but if lim sup n→∞kn/lnn > 1/lnm, then this property is not true. Also, if lim inf...kn/lnn > 1/lnm , then almost all words of length n over A do not contain the factor w. Moreover, if lim...(ln n - knln m) = a e IR, then lim sup...|W(n,kn,w,A)|/m^n < 1-exp(-exp(a)) and liminf n→∞|W(n,kn,w,A)|/m^n >1-exp(-(1-1/m)exp(a)), where W(n,kn,w,A) denotes the set of words of length n over A containing the factor w of length kn.
12
Content available remote Regexpcount, a symbolic package for counting problems on regular expressions and words
Nicodeme P.
Fundamenta Informaticae
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2003
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Vol. 56, nr 1,2
71-88
EN
In previous work [10], we considered algorithms related to the statistics of matches with words and regular expressions in texts generated by Bernoulli or Markov sources. In this work these algorithms are extended for two purposes: to determine the statistics of simultaneous counting of different motifs, and to compute the waiting time for the first match with a motif in a model which may be constrained. This extension also handles matches with errors. The package is fully implemented and gives access to high and low level commands. We also consider an example corresponding to a practical biological problem: getting the statistics for the number of matches of words of size 8 in a genome (a Markovian sequence), knowing that an (overrepresented DNA protecting) pattern named Chi occurs a given number of times.
13
Content available remote Analysis of open queueing networks using method of generating functions
Matalycki M., Gomanczuk S., Pankow A.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2002
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Vol. 1, nr 1
143--152
EN
This paper provides the method of generating functions using for calculating the time-dependent state probabilities for open queuing networks in transient regime, when the network works in conditions of peak demand.
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