iosr-Jm   Volume-3 ~ Issue-1

Abstract :The main purpose of this paper is to give the relationship between hypergeometric series and the Lucas sequence. A variety of representations in terms of finite sums and infinite series involving coefficients are obtained. While many of them are well known and some identities appear to be now.

[1] A.K.Agarwal, ``On a new kind of numbers", The Fibonacci Quarterly, Vol 28(3), ( 1990), 194-199 .
[2] G.E. Andrews, Richard Askey, Ranjan Roy ``Special Functions",Cambridge University press 1999.
[3] W.N. Bailey.`` Generalized Hypergeometric Series". Cambrige: Cambridge University Press, 1935.
[4] C.Jordan. ``Calculus of Finite Differences". New York : Chelsea, 1950.
[5] Thomas Koshy. ``Fibonacci and Lucas Numbers with Applications", 2001.
[6] E.D. Rainville. ``Special Functions", New York : Macmillan, 1967.
[7] J.Riordan. ``Cambinatorial Identities". Huntington. New York: Krieger, 1979.
[8] M.Abramowitz and I.A.Stegun. ``Handbook of Mathematical Functions". Washington, D.C.: National Bureau of standards, 1964.

Abstract:The present paper studies a new class of numbers. Results obtained in this paper are a table, recurrence relations, generating functions and Summation formulas for these new class of numbers . Many results reduce to their corresponding results for the Catalan numbers .

[1] A.K.Agarwal, ``On a new kind of numbers", The Fibonacci Quarterly, Vol 28(3), ( 1990), 194-199 .
[2] C.Jordan. ``Calculus of Finite Differences". New York : Chelsea, 1950.
[3] J.Riordan. ``Cambinatorial Identities". Huntington. New York:Krieger, 1979
[4] E.D. Rainville. ``Special Functions", New York : Macmillan, 1967.

Abstract: layer of Rivlin-Ericksen viscoelastic fluid heated from below is considered in the presence of uniform vertical magnetic field. Following the linearized stability theory and normal mode analysis, the paper mathematically established the condition for characterizing the oscillatory motions which may be neutral or unstable, for any combination of perfectly conducting, free and rigid boundaries at the top and bottom of the fluid. It is established that all non-decaying slow motions starting from rest, in a Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension and finite vertical depth, which is acted upon by uniform vertical magnetic field opposite to gravity and a constant vertical adverse free and rigid boundaries and the exact solutions of the problem investigated in closed form, are not obtainable. temperature gradient, are necessarily non-oscillatory, in the regime...........

[1] H. Bénard, Les tourbillions cellulaires dans une nappe liquid, Revue Genérale des Sciences Pures et Appliquees 11 (1900), 1261- 1271, 1309-1328.
[2] L. Rayleigh, On convective currents in a horizontal layer of fluid when the higher temperature is on the underside, Philosophical Magazine 32 (1916), 529-546.
[3] H. Jeffreys, The stability of a fluid layer heated from below, Philosophical Magazine 2 (1926), 833-844.
[4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York (1981).
[5] J.G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastic-viscous Liquids, Proceedings of the Royal Society of London A245 (1958), 278-297.
[6] R.S. Rivlin andJ.L. Ericksen, Stress deformation relations for isotropic materials, J. Rat. Mech. Anal.,4 (1955), 323.
[7] P.K. Bhatia and J.M. Steiner, Convective instability in a rotating viscoelastic fluid layer, Zeitschrift fur Angewandte Mathematik and Mechanik 52 (1972), 321-327.
[8] P.K. Bhatia and J.M. Steiner, Thermal Instability in a viscoelastic fluid layer in hydromagnetics, Journal of Mathematical Analysis and Applications 41 (1973), no. 2,271-283.
[9] R.C. Sharma, Effect of rotation on thermal instability of a viscoelastic fluid, Acta Physica Hungarica 40 (1976), 11-17.
[10] R.C. Sharma Thermal instability in a viscoelastic fluid in hydromagnetics, Acta Physica Hungarica 38 (1975), 293-298.