Some generalizations of a combinatorial identity of L. Vietoris

AbstractA remarkably simple proof is presented for an interesting generalization of a combinatorial identity given recently by L. Vietoris [Monatsh. Math. 97 (1984) 157–160]. It is also shown how this general result can be extended further to hold true for basic (or q-) series

A further note on certain dual equations involving Fourier-Laguerre series1)

AbstractThe present note aims at providing a systematic investigation of the solution of a certain pair of dual equations involving Fourier-Laguerre series. It exhibits equivalence of the solutions obtained earlier (see [5] and [6]) by using a multiplying factor technique as well as by considering separately the equations when (i) g(x)≡0, (ii) f(x)≡0, and reducing the problem in each case to that of solving an Abel integral equation.It is also shown how the solution of these dual series equations are affected when the parameters involved are constrained differently

Remarks on some series expansions associated with certain products of the incomplete Gamma functions

AbstractIn several recent works, some interesting generalizations of the first-order Volterra-type integro-differential equation governing the unsaturated behavior of the free electron laser (FEL) were introduced and investigated by making use of fractional calculus (that is, calculus of integrals and derivatives of an arbitrary real or complex order). Among other things, it is observed here that an expansion formula for the confluent hypergeometric function in a series of the product of two entire (integral) incomplete Gamma functions does not hold true as asserted and applied in these earlier works. Some necessary corrections and possible remedies are also pointed out

Some series of Fox's H-functions

AbstractIn an earlier paper H.M. Srivastava and R. Panda [J. Reine Angew. Math. 283/284, 265–274 (1976)] gave a bilateral generating function, involving the H-function of C. Fox [Trans. Amer. Math. Soc. 98, 395–429 (1961)], for a general class of hypergeometric polynomials and also considered its multivariable analogue associated with the H-function of several complex variables. The present work begins by observing, among other things, that both of the main generating functions in a paper by M. Shah [J. Reine Angew. Math. 288, 121–128 (1976)] are contained in a single bilateral generating function which was given earlier in the aforementioned paper by Srivastava and Panda [op. cit., p. 267, Eq. (2.1)]. It then proceeds to discuss several non-trivial generalizations of the various interesting consequences of the Srivastava-Panda results. Finally, it develops a simple and direct proof of an elegant generalization of a known finite series of H-functions, which happens to be the main result in another paper by Shah [J. Reine Angew. Math. 285, 1–6 (1976)], and indicates that Shah's result [op. cit., p. 3, Eq. (2.1)] would follow rather easily from Vandermonde's summation theorem.The various generalizations mentioned in the preceding paragraph are given by Theorems 1, 2 and 3 of the present paper; each of these main results is believed to be new

Origin of superconductivity in layered centrosymmetric LaNiGa2

The following article appeared in Applied Physics Letters , Vol. 104, 022603 (2014) and may be found at: http://dx.doi.org/10.1063/1.4862329We have examined the origin of superconductivity in layered centrosymmetric LaNiGa2 by employing a linear response approach based on the density functional perturbation theory. Our results indicate that this material is a conventional electron-phonon superconductor with intermediate level of coupling strength, with the electron-phonon coupling parameter of 0.70, and the superconducting critical temperature of 1.90 K in excellent accordance with experimental value of 1.97 K. The largest contribution to the electron-phonon coupling comes from the La d and Ga p electrons near the Fermi energy and the B3g phonon branch resulting from vibrations of these atoms along the Γ-Z symmetry line in the Brillouin zone. © 2014 AIP Publishing LLC

Phonon anomalies and superconductivity in the Heusler compound YPd 2Sn

The following article appeared in Journal of Applied Physics, Vol. 116, 013907 (2014) and may be found at: http://dx.doi.org/10.1063/1.4887355We have studied the structural and electronic properties of YPd 2Sn in the Heusler structure using a generalized gradient approximation of the density functional theory and the ab initio pseudopotential method. The electronic results indicate that the density of states at the Fermi level is primarily derived from Pd d states, which hybridize with Y d and Sn p states. Using our structural and electronic results, phonons and electron-phonon interactions have been studied by employing a linear response approach based on the density functional theory. Phonon anomalies have been observed for transverse acoustic branches along the [110] direction. This anomalous dispersion is merely a consequence of the strong coupling. By integrating the Eliashberg spectral function, the average electron-phonon coupling parameter is found to be λ=0.99. Using this value, the superconducting critical temperature is calculated to be 4.12K, in good accordance with the recent experimental value of 4.7K. © 2014 AIP Publishing LLC

Certain classes of series associated with the Zeta function and multiple gamma functions

AbstractThe authors apply the theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, in order to evaluate some families of series involving the Riemann Zeta function. By introducing a certain mathematical constant, they also systematically evaluate this constant and some definite integrals of the triple Gamma function. Various classes of series associated with the Zeta function are expressed in closed forms. Many of these results are also used here to compute the determinant of the Laplacian on the four-dimensional unit sphere S4 explicitly

Fractional derivatives of the H-function of several variables

AbstractIn the present paper we derive a number of key formulas involving fractional derivatives for the H-function of several variables, which was introduced and studied in a series of papers by H. M. Srivastava and R. Panda [cf., e.g., J. Reine Angew. Math. 283/284 (1976), 265–274; J. Reine Angew. Math. 288 (1976), 129–145; Comment. Math. Univ. St. Paul. 24 (1975), fasc. 2, 119–137; ibid. 25 (1976), fasc. 2, 167–197; Nederl. Akad. Wetensch. Proc. Ser. A 81 = Indag. Math. 40 (1978), 118–131 and 132–144; Nederl. Akad. Wetensch. Proc. Ser. A 82 = Indag. Math. 41 (1979), 353–362; see also Bull. Inst. Math. Acad. Sinica 9 (1981), 261–277].We make use of the generalized Leibniz rule for fractional derivatives in order to obtain one of the aforementioned results, which involves a product of two multivariable H-functions. Each of these results is shown to apply to yield interesting new results for certain multivariable hypergeometric functions and, in addition, several known results due, for example, to J. L. Lavoie, T. J. Osler and R. Tremblay [SIAM Rev. 18 (1976), 240–268], H. L. Manocha and B. L. Sharma [J. Austral. Math. Soc. 6 (1966), 470–476; J. Indian Math. Soc. (N.S.) 38 (1974), 371–382] and R. K. Raina and C. L. Koul [Jñānābha 7 (1977), 97–105]

A certain class of rapidly convergent series representations for ζ(2n+1)

AbstractFor a natural number n, the authors propose and develop three new series representations for the Riemann Zeta function ζ(2n+1). The infinite series occurring in each of these three representations for ζ(2n+1) converges remarkably faster than that in Wilton's result. Furthermore, one of the three series representations for ζ(2n+1) involves the most rapidly convergent series among all the hitherto known members of the family of series representations considered here. Relevant connections of the results presented in this paper with many other known series representations for ζ(2n+1) are also briefly indicated

Fractional integration of the H-function of several variables

AbstractThe main object of the present paper is to derive a number of key formulas for the fractional integration of the multivariable H-function (which is defined by a multiple contour integral of Mellin-Barnes type). Each of the general Eulerian integral formulas (obtained in this paper) are shown to yield interesting new results for various families of generalized hypergeometric functions of several variables. Some of these applications of the key formulas would provide potentially useful generalizations of known results in the theory of fractional calculus