h asymptotic stability by fixed point in neutral nonlinear differential equations with delay abstract we use fixed point theory to obtain h asymptotic stability results about the zero solution of nonlinear differential equations with functional delay neither boundedness of the deviations nor a fixed sign on the coefficients functions are asked an explicit estimation of the solutions is obtained and a necessary and sufficient condition is determined classical results are improved and generalized

variationally stable differential systems let d e ir be a domain 0 e d and z a cc where a is a fixed real number let f c z x d if f t 0 0 consider the nonlinear system x f t x 1 and the perturbation g e c z x d rn such that for suitable functions ai 0 ii e c z i 1 p oi e c 0 co i 1 p nondecreasing and positive functions on 0 00 such that l w l o 11 and o oi are nondecreasing functions on 0 co we study the stability of the perturbed system y f t y g c yl knowing the stability of the variational systems 3 z fx t x t to xo z 4 where x t to x0 is the solution of 1 passing through the point to x0 e z x d thus we prove that if system 4 hence 1 is uniformly lipschitz stable see 3 then 3 is uniformly stable see theorem l and if q u uai clib l

radon cosine and sine transforms on real hyperbolic space new pointwise inversion formulae are obtained for the d dimensional totally geodesic radon transform on the n dimensional real hyperbolic space 1 d n 1 in terms of polynomials of the laplace beltrami operator and intertwining fractional integrals similar results are established for hyperbolic cosine and sine transforms

series representations for some mathematical constants abstract the authors apply a classical series identity involving the psi or digamma function with a view to deriving series representations for a number of known mathematical constants several closely related consequences and results are also considered

the h function associated with a certain class of feynman integrals motivated by some further examples of the use of feynman integrals which arise in perturbation calculations of the equilibrium properties of a magnetic model of phase transitions a generalization of the familiar h function of fox 1961 was proposed recently by inayat hussain 1987 a brief discussion of contour selection and convergence conditions of the integrals involved is presented it is also shown how some recent work of buschman 1990 can be applied to derive various recurrence relations for the general h function finally some relevant comments are made on the validity and novelty of two summation formulae for the clausenian hypergeometric function which were derived by inayat hussain

some classes of infinite series associated with the riemann zeta and polygamma functions and generalized harmonic numbers the authors aim at presenting a systematic investigation of several families of infinite series which are associated with the riemann zeta function the digamma and polygamma functions the harmonic and generalized harmonic numbers and the stirling numbers of the first kind relevant connections of the results derived here with those considered in many earlier works on this subject are also indicated

fractional derivatives of certain generalized hypergeometric functions of several variables abstract the main object of the present paper is to derive a number of key formulas for the fractional derivatives of the multivariable h function which is defined by a multiple contour integral of mellin barnes type each of these formulas can be shown to yield interesting new results for various classes of generalized hypergeometric functions of several variables some of these applications of the key formulas provide potentially useful generalizations of known results in the theory of fractional calculus

a certain family of infinite series associated with digamma functions abstract the sums of several interesting infinite series were recently expressed in terms of the psi or digamma functions the object of this paper is to present a systematic account of these and of numerous similar or more general series whose sums can be found in the literature in various equivalent forms some relevant unifications and further generalizations are also indicated

multiplication of fractional calculus operators and boundary value problems involving the euler darboux equation abstract with a view to presenting solutions of various boundary value problems involving the celebrated euler darboux equation the authors consider the multiplication of certain classes of operators of fractional calculus defined in terms of the gaussian hypergeometric function the fractional calculus operators studied here incorporate as their special cases both the riemann liouville and erdelyi kober operators and are appropriately restricted in order to yield explicit solutions of some of the aforementioned boundary value problems in terms of appell functions and kampe de feriet functions of two variables

some multidimensional fractional integral operators involving a general class of polynomials in this paper the authors present a systematic investigation of two novel families of multidimensional fractional integral operators involving a general class of polynomials with essentially arbitrary coefficients the various theorems established here involve compositions inversion formulas and multidimensional mellin transforms and convolutions of these general fractional integral operators each of the results obtained in this paper would unify and extend the corresponding known or new result for simpler families of fractional integral operators which were studied in several earlier works on the subject

generalized watson s summation formula for 3 f 2 1 abstract a summation formula is given for 3 f 2 a b c 1 2 a b i 1 2c j 1 with fixed j and arbitrary i i j z this result generalizes the classical watson s theorem which deals with the case i j 0 extensions to the cases of 3f2 a 1 i j a c e 1 i 2c e 1 and 3f2 a b c 1 i a b 1 i j a c 1 are given notice that the case i j 0 corresponds to the classical theorems due to whipple and dixon respectively

evaluation of a certain class of eulerian integrals in a study of the screening properties of a charged impurity located inside and near the surface of a metal subjected to a magnetic field there arose an interesting class of eulerian integrals involving the bessel function jv z or j0 z which were expressed in closed forms by glasser 1984 subsequently wille 1988 generalized and extended glasser s work by giving closed form expressions for a number of eulerian integrals involving meijer s g function motivated by these works the authors aim at evaluating a general class of eulerian integrals involving fox s h function their main result is shown to provide a key formula from which one can deduce each of the aforementioned integrals and also numerous other potentially useful results not contained in the earlier works they are led in this way to the corrected version of one of wille s main integral formulae

some subclasses of analytic functions associated with fractional calculus operators abstract this paper begins by studying several properties and characteristics of certain subclasses of analytic functions with positive coefficients the results investigated here include various coefficient inequalities distortion properties and the radii of close to convexity inclusion theorems involving the hardy space of analytic functions and the class of functions whose derivative has a positive real part are also investigated relationships between certain subclasses of analytic functions involving fractional derivative operators with negative coefficients and a certain generalized fractional integral operator are then studied various known or new special cases of our results are also pointed out

some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients abstract the object of the present paper is to prove various distortion theorems for the fractional calculus of functions in the subclasses r and c consisting of prestarlike and normalized analytic functions with negative coefficients furthermore distortion theorems involving a generalized fractional integral operator for functions in the subclasses r and c are given

wave propagation and thom s theorem scalar waves propagated from a finite emitter in an inhomogeneous medium form caustics whose geometry and location are functions of both the boundary conditions on the emitter and the inhomogeneity of the medium the wave amplitude satisfies a helmholtz equation for the class of problems considered a phase is derived from a characterisation of the medium and the boundary conditions on the emitter the location and geometry of the caustic in configuration space is determined from phase space considerations through the lagrange manifold an oscillatory integral whose asymptotic expansion is that of the amplitude is determined by a sharpening of maslov s method 1972 of characteristics thom s theorem prescribes a normal form for the phase function in the integral transformations carrying the phase function to the canonical form are determined then following duistermaat 1973 the complete asymptotic series of the integral and hence of the amplitude is obtained the entire algorithm is illustrated with a specific example

a certain class of infinite sums associated with digamma functions the sums of several interesting families of infinite series were recently expressed in terms of the psi or digamma functions the main object of this paper is to present a further generalization of two such classes of infinite sums which were considered earlier by al saqabi et al j math anal appl 159 1991 361 372 some other results relevant to the present investigation are also indicated

a subclass of analytic functions defined by using certain operators of fractional calculus making use of certain operators of fractional calculus we introduce a new class f6 n a a of functions which are analytic in the open unit disk 4 and obtain a necessary and sufficient condition for a function to be in the class fs n a ct we also determine the radii of close to convexity starlikenees and convexity finally an application involving fractional calculus of functions in the class f6 n a a is considered

some infinite series and functional relations that arose in the context of fractional calculus abstract several interesting infinite series relations were derived recently by applying such operators of fractional calculus as the familiar riemann liouville fractional differintegral operator 0 d z of real or complex order the main object of this paper is to present much simpler alternative derivations of substantially more general families of infinite series relations without using fractional calculus some relevant connections among various known results are also provided

some summation formulas for the series 3f2 1 summation formulas contiguous to watson s and whipple s theorems in the theory of the generalized hypergeometric series are obtained certain limiting cases of these results are given 1 two results contiguous to watson s theorem the two following summation formulas for series 3f2 1 are useful interesting easily established and probably new they are

h k trichotomies and asymptotics of nonautonomous difference systems abstract discrete nonautonomous nonlinear systems possessing h k trichotomies are considered we specialize to the case of solutions moving inside and in a neighborhood of an invariant manifold of a special kind which we call h k hyperbolic the aulbach coppel knobloch transformation is then developed and used to state conditions for granting asymptotic equivalence of solutions our results generalize some of aulbach s results in which the manifold considered was made of stationary points and the system taken was autonomous

generalizations of dixon s theorem on the sum of a 3 f 2 twenty three formulas closely related to dixon s theorem in the theory of the generalized hypergeometric series are obtained twenty six lim iting cases are also deduced

lipschitz stability of nonlinear systems of differential equations in s the authors introduced the notion of lipschitz stability in differential equations this notion lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation 3 and uniform stability in variation 4 on the other side however lipschitz stability is new only as a nonliner phenomenon since it coincides with uniform stability in linear systems s an important feature of lipschitz stability is that unlike uniform stability the linearized system inherits the property of lipschitz stability from the original nonlinear system 5 in this paper we pursue the study of lipschitz stability that started in s using essentially the techniques of liapunov functions then we give sufficient conditions for the lipschitz stability of certain nonlinearly pertur bed nonlinear systems such systems include among other equations certain integrodifferential and functional differential equations then we give an example which can be investigated successfully using our results but cannot be handled by any previous techniques or results 9

on an elastodynamic classification of orthorhombic media sets of elastic constants for crystalline media are normally presented with reference to the axes of the unit cell defined by structural crystallographers landolt bornstein 1979 in orthorhombic materials the magnitudes of the nine elastic stiffnesses in general show little or no consistent correspondence with the suffixes related to crystal axes of the physical constants a classification scheme for orthorhombic media based on an ordering of the elastodynamically important principal stiffnesses of extension c 11 c 22 c 33 and of shear c 44 c 55 c 66 is described and illustrated the classification facilitates meaningful comparisons of elastodynamic properties since it offers the following advantages i it is amenable to cyclic interchange ii it permits concise expression of basic homogeneous plane wave theory musgrave 1970 iii it affords rapid appreciation of the possible essential configurations of homogeneous plane wave slowness in each coordinate plane iv it allows known criteria for the existence of hyperbolic regions in elastic slowness and wave surfaces musgrave 1957 1979 and some further developments to be succinctly expressed application of the scheme to the theory of bulk waves is exemplified in detail for three media diverse in their properties alpha uranium spruce wood and a hypothetical medium discussed by al shits lothe 1979b specification of exceptional limiting bulk waves chadwick smith 1977 is included

hypergeometric identities arising from the elephant random walk abstract a probabilistic approach is provided to establish new hypergeometric identities it is based on the calculation of moments of the limiting distribution of the position of the elephant random walk in the superdiffusive regime

fundamental elastodynamic solutions for anisotropic media with ellipsoidal slowness surfaces when the slowness surface ell of an anisotropic elastic medium consists of three concentric ellipsoids solutions of the displacement equations of motion can be generated from functions satisfying scalar wave equations and the problem of constructing the fundamental or green s tensor g for an infinite region becomes tractable this paper has two aims first to find all the conditions on the linear elastic moduli under which ell is ellipsoidal that is the union of concentric ellipsoids and second to determine g for each case in which scr g simplifies in this way the two stages of the investigation have a key idea in common the ellipsoidal form of scr g requires the eigenvalues of the acoustical tensor q n to be quadratic forms in the unit vector argument n at least two of the associated eigenvectors are either constant or linear in n and the squared moduli of the linear eigenvectors are divisors of eigenvalue differences these algebraic properties provide a classification of media with ellipsoidal slowness surfaces and aid in characterizing the membership of each class the first stage culminates in four sets of conditions labelled a b c i and c ii case c i is a restriction of transverse isotropy and the others are specializations of orthorhombic symmetry at the second stage n is replaced by the gradient partial with respect to spatial position and polynomials in n become differential operators the construction of g involves two canonical problems of classical type an initial value problem for a scalar wave equation and a potential problem for a pair of charged ellipsoids the divisibility property indicated above implies that the ellipsoids are confocals carrying equal and opposite charges and these characteristics render the fundamental solution causal in the sense that the entire disturbance excited by the point impulse begins with the first and ends with the last of the wavefront arrivals the structures of the fundamental solutions in cases a b c i and c ii are described and the latter solution is shown to reduce to a standard result of stokes in the degenerate case of isotropy mention is also made of a specialization of case b appropriate to a transversely isotropic medium which is inextensible in the direction of the symmetry axis

asymptotic relation for the transition density of the three dimensional markov random flight on small time intervals we consider the markov random flight varvec x t t 0 in the three dimensional euclidean space mathbb r 3 with constant finite speed c 0 and the uniform choice of the initial and each new direction at random time instants that form a homogeneous poisson flow of rate lambda 0 series representations for the conditional characteristic functions of varvec x t corresponding to two and three changes of direction are obtained based on these results an asymptotic formula as t rightarrow 0 for the unconditional characteristic function of varvec x t is derived by inverting it we obtain an asymptotic relation for the transition density of the process we show that the error in this formula has the order o t 3 and therefore it gives a good approximation on small time intervals whose lengths depend on lambda an asymptotic formula as t rightarrow 0 for the probability of being in a three dimensional ball of radius r ct is also derived estimate of the accuracy of the approximation is analysed

summations for basic hypergeometric series involving a q analogue of the digamma function using a simple method numerous summation formulas for hypergeometric and basic hypergeometric series are derived among these summation formulas are nonterminating extensions and q extensions of identities recorded by lavoie luke watson and srivastava at the result side of the basic hypergeometric summations there appears a q analogue of the digamma function some of its properties are also studied

some characterization theorems for starlike and convex functions involving a certain fractional integral operator abstract the present paper is devoted to the investigation of the sufficient conditions that are satisfied by the generalized fractional integrals of certain analytic functions in the open unit disk in order to be starlike or convex further characterization theorems involving the hadamard product or convolution are also considered

fractional calculus and the sums of certain families of infinite series abstract although the concept of fractional calculus can be traced back to l hopital s naive three centuries old curiosity about the meaning of leibniz s nth derivative notation dny dxn when n 1 2 and many monumental works on the subject of fractional calculus have since appeared and are still appearing in the mathematical literature the five volume work published recently by k nishimoto fractional calculus vols i iv 1984 1987 1989 1991 an essence of nishimoto s fractional calculus calculus in the 21st century integrations and differentiations of arbitrary order 1991 contains an interesting account of the theory and applications of fractional calculus in a number of areas of mathematical analysis such as ordinary and partial differential equations summation of series et cetera the main object of the present paper is to examine rather systematically some of the most recent contributions by nishimoto et al cf e g j college engrg nihon univ ser b 32 1991 7 13 15 21 23 30 j fractional calculus1 1992 7 16 see also fractional calculus vol iv 1991 j college engrg nihon univ ser b33 1992 1 8 l galue et al j fractional calculus 1 1992 17 21 and others on the use of fractional calculus in finding the sums of numerous interesting families of infinite series various other classes of infinite sums found in the literature without using fractional calculus and their hitherto unnoticed connections with certain known results are also considered

some fractional differintegral equations recently nishimoto j college engrg nihon univ ser b 24 1983 7 13 studied a certain fractional integral equation and a system of simultaneous differintegral equations each of order 12 motivated by nishimoto s work owa and nishimoto j college engrg nihon univ ser b 24 1983 67 72 considered a general fractional differintegral equation the object of the present paper is to investigate some interesting properties of functions which satisfy the general fractional differintegral equation solved by owa and nishimoto

some applications of fractional integral operators and ruscheweyh derivatives abstract recently owa saigo and srivastava introduced and studied a certain generalized fractional integral operatorj0 z involving the gaussian hypergeometric function the object of this paper is to investigate various properties and relationships involvingj0 z the carlson shaffer operator l a c and the ruscheweyh derivatived a number of interesting subclasses of analytic and univalent functions are also considered in our investigation

a calculus for finely layered anisotropic media matrix algebra and group theory combine to offer a formalism for the simple calculation of the elastic anisotropic homogeneous medium which is equivalent in the long wavelength limit to a heterogeneous distribution of fine layers each layer itself an elastic anisotropic medium the properties of each anisotropic constituent in a set of fine layers map to an element of a commutative group a reverse mapping returns the material properties of the constituent adding group elements gives the group element for the homogeneous medium equivalent to a heterogeneous set of layers addition of an inverse element that is subtraction provides the means to remove a set of layers from an anisotropic medium then if the remaining layer is a stable anisotropic medium a valid decomposition of the original medium into anisotropic constituents is obtained within the group structure eight subgroups corresponding to eight types of elastic symmetry systems may be identified immediately yielding the symmetry of the e

a certain subclass of analytic functions associated with operators of fractional calculus abstract in the present paper we introduce a new class of functions defined by using certain fractional calculus operators which are analytic in the open unit disk a necessary and sufficient condition for a function to belong to such a class is obtained the various other results presented here include the radii of close to convexity starlikeness and convexity and some distortion theorems involving fractional integrals and fractional derivatives several special cases of our results are also pointed out

a simplified overview of certain relations among infinite series that arose in the context of fractional calculus abstract an interesting infinite series relation was proven recently by applying a certain known generalization of the riemann liouville and erdelyi kober operators of fractional calculus in our attempt to give a much simpler proof of this result without using the generalized fractional calculus operator we are led here to another class of infinite series relations some relevant connections with a number of known results are also indicated

asymptotic behavior of nonlinear difference systems in this paper we study a general variational stability introduced mainly for weakly stable difference systems moreover we obtain asymptotic formulae for these systems which state new results about asymptotic behavior for perturbed systems under general hypotheses

fractional integral operators involving a general class of polynomials abstract in the present paper the authors derive a number of interesting expressions for the composition of certain multidimensional fractional integral operators involving a general class of polynomials with essentially arbitrary coefficients it is shown how these fractional integral operators can be identified with elements of the algebra of functions having the multidimensional mellin convolution as the product inversion formulas for the multidimensional fractional integrals are also established the fractional integral operators studied here are fairly general in character since by suitably specializing the coefficients involved the general class of polynomials can be reduced to each of the classical orthogonal polynomials the bessel polynomials and numerous other classes of generalized hypergeometric polynomials studied in the literature

asymptotic integration of a system resulting from the perturbation of an h system z systems were introduced in lo with the intention of obtaining results about stability for a weakly stable system at least weaker than those given by exponential stability under some perturbations in this paper we get asymptotic formulae for these systems which state new results about asymptotic integration for nonlinear systems under general hypotheses moreover the corresponding results for linear systems give new insight about levinson s theorem on asymptotic integration in this way using the h systems we get a uniform treatment for the concept of stability and we verify that several classical theorems of stability take a precise form by the asymptotic formulae in section 2 we recall the definition and some properties of an h system and we state two new theorems about asymptotic behavior of solutions of perturbed h systems in section 3 some general results about asymptotic integration are given these results extend those given in lo in section 4 we apply the technique to the linear case finally in section 5 we show examples and applications and we study the emden fowler equation which originally does not satisfy the general hypothesis but can be transformed into one where they are satisfied 2 h systems let us consider the differential system

integral inequalities of gronwall bellman bihari type and asymptotic behavior of certain second order nonlinear differential equations abstract several integral inequalities similar to gronwall bellmann bihari inequalities are obtained these inequalities are used to discuss the asymptotic behavior of certain second order nonlinear differential equations

the geometry of the hyperbolic system for an anisotropic perfectly elastic medium we evaluate the fundamental solution of the hyperbolic system describing the generation and propagation of elastic waves in an anisotropic solid by studying the homology of the algebraic hypersurface defined by the characteristic equation also known as the slowness surface our starting point is the herglotz petrovsky leray integral representation of the fundamental solution we find an explicit decomposition of the latter solution into integrals over vanishing cycles associated with the isolated singularities on the slowness surface as is well known in the theory of isolated singularities integrals over vanishing cycles satisfy a system of differential equations known as picard fuchs equations such equations are linear and can have at most regular singular points we discuss a method to obtain these equations explicitly subsequently we use the monodromy properties around the regular singular points to find the asymptotic behavior according to the different types of singularities that may appear on a wave front in three dimensions this is a method alternative to the one that arises in the maslov theory of oscillating integrals our work sheds new light on how to compute and classify the cagniard de hoop contour in the complex radial horizontal slowness plane this contour is used in numerical integration schemes to obtain the full time behaviour of the fundamental solution for a given direction of propagation

a linear fractional calculus operator and its applications to certain subclasses of analytic functions for a real number a 1 2 3 we define by using fractional calculus a linear operator a a on the class 4 of analytic functionsf z in the open unit disk z 1 satisfying the usual normalization conditions f 0 0 and f o 1 this operator is related to the linear operator a c introduced by b c carlson and d b shaffer l in their study of starlike and prestarlike hypergeometric functions we prove several interesting starlike and convex properties of the function a a f z

commutativity of the leibniz rules in fractional calculus abstract many earlier works on the subject of fractional calculus that is differentiation and integration of an arbitrary real or complex order provide interesting accounts of the theory and applications of fractional calculus operators in several areas of mathematical analysis such as ordinary and partial differential equations integral equations special functions summation of series etc the main object of this sequel to the aforementioned works is to examine rather closely the commutativity of the familiar leibniz rules for fractional calculus and its various consequences some generalizations of a recent result of tu chyan and wu 1 involving fractional integration of powers of the logarithmic functions are also considered

some evaluations for the generalized hypergeometric series whipple s theorem on the sum of a 3 f2 1 plays a key role in obtaining a family of summation formulas for the generalized hypergeometric series of unit argument 1 introduction and main result the object of this paper is to put on record a family of evaluation formulas for the generalized hypergeometric series of unit argument in terms of the logarithmic derivative of the gamma function z d ln f z dz and the polygamma function 4 z d 4i z dzn these results are

operators of fractional integration and their applications the main purpose of this paper is to present a systematic and historical account of the investigations carried out by various authors in the field of fractional calculus and its applications several interesting results relevant to the present investigation are also considered

a theorem relating a certain generalized weyl fractional integral with the laplace transform and a class of whittaker transforms abstract in the present paper the authors prove a theorem which asserts an interesting relationship between the classical laplace transform a certain class of whittaker transforms and a weyl fractional integral involving a general class of polynomials with essentially arbitrary coefficients by specializing the various parameters involved this general theorem would readily yield several known or new results involving simpler integral operators it is also shown how the relationship asserted by the theorem can be applied to evaluate the generalized weyl fractional integrals of various special functions

a class of distortion theorems involving certain operators of fractional calculus abstract the object of the present paper is to investigate a general class of fractional integral operators involving the gauss hypergeometric function several interesting distortion theorems for various subclasses of analytic and univalent functions are proved in terms of these operators of fractional calculus some special cases of the results presented here are also indicated

the exact solution of certain differential equations of fractional order by using operational calculus abstract in the present paper the authors first develop an operational calculus for the familiar riemann liouville fractional differential operator this operational calculus is then used here to solve a cauchy boundary value problem for a certain linear equation involving the riemann liouville fractional derivatives relevant connections are also indicated with the special cases of the equation which were solved earlier by using other methods

generalizations of whipple s theorem on the sum of a 3 f 2 thirty eight summation closely related to whipple s theorem in the theory of the generalized hypergeometric series are obtained some limiting cases are also considered

some transformation and reduction formulas for hypergeometric series in several variables abstract an alternative elementary proof of a certain hypergeometric identity which was posed recently as a problem leads naturally first to a multiple series extension of the problem and then also to its related multivariable hypergeometric transformation and summation formulas a systematic analysis of several general multiple series identities analogous to these hypergeometric transformations is presented many of the general results obtained are shown to unify and extend numerous transformation and reduction formulas for various classes of double and multiple hypergeometric series

fractional derivatives of the h function of several variables in 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 jn n bha 7 1977 97 105

fractional calculus operators and their applications involving power functions and summation of series many earlier works on the subject of fractional calculus contain interesting accounts of the theory and applications of fractional calculus operators in a number of areas of mathematical analysis such as ordinary and partial differential equations integral equations summation of series etc the main object of this paper is to examine rather systematically and extensively some of the most recent contributions on the applications of fractional calculus operators involving power functions and in finding the sums of several interesting families of infinite series various other classes of infinite sums found in the mathematical literature by these or other means and their validity or hitherto unnoticed connections with some known results are also considered

on h k manifolds with asymptotic phase abstract we introduce the notion of h k manifolds and give conditions under which the property of being a manifold with asymptotic phase holds these conditions allow the construction of a transformation of the variational equation of a nonlinear nonautonomous systems with h k dichotomies into an almost decoupled quasilinear system the class of admissiblehandkfunctions is also briefly analyzed