# Search Results

###### Oscillation and Periodicity of a Second Order Impulsive Delay Differential Equation with a Piecewise Constant Argument

References [1] M. Akhmet: Nonlinear hybrid continuous/discrete-time models.. Springer Science & Business Media (2011). [2] H. Bereketoglu, G.S. Oztepe: Convergence of the solution of an impulsive differential equation with piecewise constant arguments. Miskolc Math. Notes 14 (2013) 801-815. [3] H. Bereketoglu, G.S. Oztepe: Asymptotic constancy for impulsive differential equations with piecewise constant argument. Bull. Math. Soc. Sci. Math. Roumanie Tome 57 (2014) 181-192. [4] H. Bereketoglu, G

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Generalized Jacobsthal numbers and restricted *k*-ary words

## Abstract

We consider a generalization of the problem of counting ternary words of a given length which was recently investigated by Koshy and Grimaldi [10]. In particular, we use finite automata and ordinary generating functions in deriving a *k*-ary generalization. This approach allows us to obtain a general setting in which to study this problem over a *k*-ary language. The corresponding class of *n*-letter *k*-ary words is seen to be equinumerous with the closed walks of length *n* − 1 on the complete graph for *k* vertices as well as a restricted subset of colored square-and-domino tilings of the same length. A further polynomial extension of the *k*-ary case is introduced and its basic properties deduced. As a consequence, one obtains some apparently new binomial-type identities via a combinatorial argument.

###### On the notion of Jacobi fields in constrained calculus of variations

## Abstract

In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of local gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' strengths [16]. In dis- cussing the positivity of the second variation, a relevant role is played by the Jacobi fields, defined as infinitesimal generators of 1-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this point is presented. An alternative characterization of the Jacobi fields as solutions of a suitable accessory variational problem is established.

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Approach of *q*-Derivative Operators to Terminating *q*-Series Formulae

–271. [28] Z. Liu: An expansion formula for q -series and applications. Ramanujan J. 6 (4) (2002) 429–447. [29] B.M. Minton: Generalized hypergeometric function of unit argument. J. Math. Phys. 11 (4) (1970) 1375–1376. [30] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/17.7 [31] I.J. Slater: Generalized Hypergeometric Functions . Cambridge University Press, Cambridge (1966). [32] A. Verma, V.K. Jain: Some summation formulae for nonterminating basic hypergeometric series. SIAM J. Math. Anal. 16 (3) (1985) 647

###### Integrals of logarithmic and hypergeometric functions

References [1] V. Adamchik, H. M. Srivastava: Some series of the zeta and related functions. Analysis 18 (2) (1998) 131-144. [2] J. M. Borwein, I. J. Zucker, J. Boersma: The evaluation of character Euler double sums. Ramanujan J. 15 (2008) 377-405. [3] J. Choi: Log-Sine and Log-Cosine Integrals. Honam Mathematical J 35 (2) (2013) 137-146. [4] J. Choi, D. Cvijović: Values of the polygamma functions at rational arguments. J. Phys. A: Math. Theor. 40 (50) (2007) 15019{15028. Corrigendum, ibidem, 43

###### Combinatorial proofs of some Stirling number formulas

## Abstract

In this note, we provide bijective proofs of some recent identities involving Stirling numbers of the second kind, as previously requested. Our arguments also yield generalizations in terms of a well known q-Stirling number.