multiplicity results for asymptotically linear elliptic problems at resonance in this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via morse theory and minimax methods some new observations on the critical groups of a local linking type critical point are used to deal with the resonance case at 0

cubic theta functions some new identities for the four cubic theta functions a q z a q z b q z and c q z are given for example we show that a q z 3 b q z 3 c q 2c q z this is a counterpart of the identity a q z 3 b q 2b q z3 c q z 3 which was found by hirschhorn et al the laurent series expansions of the four cubic theta functions are given their transformation properties are established using an elementary approach due to k venkatachaliengar by applying the modular transformation to the identities given by hirschhorn et al several new identities in which a q z plays the role of a q z are obtained

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

monotonicity properties of generalized elliptic integrals with respect to the parameter abstract for a 0 1 2 and r 0 1 let k a r and e a r k r and e r denote the generalized elliptic integrals the complete elliptic integrals respectively of the first and second kinds respectively in this paper the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a 0 1 2 by these results sharp bounds expressed in terms of k r e r and elementary functions are obtained for k a r and e a r thus showing the dependence on the parameter a of k a r and e a r and substantially improving the known related results

some properties of the difference between the ramanujan constant and beta function abstract the authors present the power series expansions of the function r a b a at a 0 and at a 1 2 show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so called ramanujan constant r a and the beta function b a b a 1 a and obtain asymptotically sharp lower and upper bounds for r a in terms of b a and polynomials in addition some properties of the riemann zeta function n n n and its related sums are derived

application of mahler measure theory to the face centred cubic lattice green function at the origin and its associated logarithmic integral the mathematical properties of the face centred cubic lattice green function and the associated logarithmic integral are investigated where c cos and w w1 iw2 is a complex variable in the w plane in particular the theory of mahler measures is used to obtain a closed form expression for s w in terms of 5f4 generalized hypergeometric functions the method of analytic continuation is then applied to this result in order to prove that next the relation ds dw g w where w 3 is used to derive the simple formula where w lies in a restricted region of the cut w plane the limit function is also evaluated in the intervals w1 1 0 and w1 0 3 it is shown that gr w1 and gi w1 can be expressed in terms of 2f1 z w1 hypergeometric functions where the independent variable z w1 is a real valued rational function of w1 finally new formulae are derived for the number of random walks rn on the face centred cubic lattice which return to their starting point not necessarily for the first time after n nearest neighbour steps

class number three ramanujan type series for 1 p abstract there is a class of remarkable series for 1 of the form c 3 n 0 a nb c 3n 6n 3n n 3 where a b c are certain algebraic numbers these were first examined by ramanujan these examples arise by computing singular invariants for j and the constants involved will have degree equal to the class number of the associated imaginary quadratic field our intention is compute all such class number three examples the largest such example with discriminant 907 adds 37 additional digit accuracy per term a class number four example with discriminant 1555 gives 50 digits per term

convexity with respect to holder mean involving zero balanced hypergeometric functions in this note we investigate the convexity of zero balanced hypergeometric functions with respect to holder mean

ramanujan s inversion formulas for the lemniscate and allied functions abstract in the unorganized portions of his second notebook ramanujan states without proofs 10 inversion formulas for the lemniscate integral and two similar integrals these 10 formulas are proven here

functional inequalities for the quotients of hypergeometric functions letf a b c x be the gaussian hypergeometric series and for 0 r 1 let formula g d anderson m k vamanamurthy and m vuorinen raised recently the following problem for whicha b 0 1 does formula hold for allr s 0 1 they also proved this inequality fora b 1 2 the main purpose of this paper is to give an answer to this problem fora b 1 and to find a lower bound for the summ r m s fora 0 2 andb 0 2 a

existence of positive solutions for a semipositone boundary value problem on the half line in this paper we consider the boundary value problem on the half line x t k 2 t x t lm t f t x t 0 t 0 x 0 0 limt x t 0 where k 0 0 and f 0 x 0 r are continuous we show the existence of positive solutions by using a fixed point theorem in cones

solution to a conjecture on the legendre m function with an application to the generalized modulus abstract in this note we give a solution to a conjecture on the legendre m function proposed by heikkala linden vamanamurthy and vuorinen in the paper heikkala et al 2008 5 as an application of this result we obtain a functional inequality of the generalized modulus

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

positive solutions of second order two delay differential systems with twin parameter abstract the paper is concerned with the existence of positive solutions for a second order two delay differential systems with twin parameter firstly we present some sufficient conditions with the twin parameter belonging to an open rectangular region eigenvalue region to ensure the existence of positive solutions for the differential systems with singularity secondly we deal with the superlinear semipositone problem of the differential systems and establish the existence of positive solutions for the systems the proofs are based on a fixed point theorem in cones

some qualitative results of the critical groups for the p laplacian equations abstract in this paper some qualitative results of the critical groups of an isolated critical point for the p laplacian equation and the existence of sign changing solutions of this equation are given by morse theory

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

existence and multiplicity for a resonance problem for the p laplacian on bounded domains in rn abstract the existence of at least two solutions for a resonance problem involving the p laplacian is shown for the case of bounded domains in r n this work constitutes an extension of a previous result of landesman robinson and rumbos for the case p 2 e landesman s robinson a rumbos multiple solutions of semilinear elliptic problems at resonance nonlinear analysis tma 24 7 1995 1049 1059

basic representations for eisenstein series from their differential equations abstract in this paper we provide a new approach for the derivation of parameterizations for the eisenstein series we demonstrate that a variety of classical formulas may be derived in an elementary way without knowledge of the inversion formulae for the corresponding schwarzian triangle functions in particular we provide a new derivation for the parametric representations of the eisenstein series in terms of the complete elliptic integral of the first kind the proof given here is distinguished from existing elementary proofs in that we do not employ the jacobi ramanujan inversion formula relating theta functions and hypergeometric series our alternative approach is based on a lie symmetry group for the differential equations satisfied by certain eisenstein series we employ similar arguments to obtain parameterizations from ramanujan s alternative signatures and those associated with the inversion formula for the modular j function moreover we show that these parameterizations represent the only possible signatures under a certain assumed form for the lie group parameters

some cubic modular identities of ramanujan there is a beautiful cubic analogue of jacobi s fundamental theta function identity it is sum n m infty infty q n 2 nm m 2 sum n m infty infty n m q n nm m sum n m infty infty q n 1 3 n 1 3 m 1 3 m 1 3 here exp 2 i 3 in this note we provide an elementary proof of this identity and of a related identity due to ramanujan we also indicate how to discover and prove such identities symbolically

a combinatorial proof of the farkas kra theta function identities and their generalizations abstract recently farkas acid kra found some cubic theta function identities from their work on automorphic forms shortly thereafter farkas and ropeliovich were able to generalize these to p th order theta function identities using the theory of elliptic functions we give short elementary proofs of the cubic identities we show that the p th order identities follow from more general relations between the coefficients of certain theta functions our proof is combinatorial and utilizes certain orthogonal transformations

functional inequalities involving special functions ii this paper is motivated by an open problem of anderson vamanamurthy and vuorinen we prove a chain of inequalities for hypergeometric functions generalized and normalized bessel functions kummer functions and for some general power series some particular cases and refinements are given

applications of morse theory to some nonlinear elliptic equations with resonance at zero abstract in this paper we study the existence and multiplicity of solutions for some nonlinear elliptic boundary value problems with resonance at zero by applying morse theory we do not impose additional global condition on the nonlinearities except for a subcritical growth condition

concavity with respect to holder means involving the generalized grotzsch function the generalized grotzsch function has numerous applications in geometric function theory and analytic number theory and its properties have been investigated by many authors in this paper we study the concavity of the generalized grotzsch function with respect to holder means

existence and multiplicity results for dirichlet problems with p laplacian in this paper we apply morse theory to study the existence of nontrivial solutions of p laplacian type dirichlet boundary value problems

nontrivial critical points for asymptotically quadratic function abstract in this paper we study the existence of nontrivial critical points as applications we study the existence of nontrivial solutions of the dirichelet problem for nonlinear elliptic pde and periodic boundary value problem for nonlinear ope

an addition formula for the jacobian theta function and its applications in this paper we prove an addition formula for the jacobian theta function using the theory of elliptic functions it turns out to be a fundamental identity in the theory of theta functions and elliptic function and unifies many important results about theta functions and elliptic functions from this identity we can derive the ramanujan cubic theta function identity winquist s identity a theta function identities with five parameters and many other interesting theta function identities and all of which are as striking as winquist s identity this identity allows us to give a new proof of the addition formula for the weierstrass sigma function a new identity about the ramanujan cubic elliptic function is given the proofs are self contained and elementary

functional inequalities for modified struve functions in this paper by using a general result on the monotonicity of quotients of power series our aim is to prove some monotonicity and convexity results for the modified struve functions moreover as consequences of the above mentioned results we present some functional inequalities as well as lower and upper bounds for modified struve functions our main results complement and improve the results of joshi and nalwaya

on sign changing and multiple solutions of the p laplacian abstract in this paper we construct the pseudo gradient vector field in w 0 1 p by which we obtain the positive and negative cones of w 0 1 p are both invariant sets of the descent flow of the corresponding functional then we use differential equations theory in banach spaces and dynamics theory to study p laplacian boundary value problems with jumping nonlinearities at zero or infinity and get new multiple solutions and sign changing solutions theorems of p laplacian

elliott s identity and hypergeometric functions elliott s identity involving the gaussian hypergeometric series contains as a special case the classical legendre identity for complete elliptic integrals the aim of this paper is to derive a differentiation formula for an expression involving the gaussian hypergeometric series which for appropriate values of the parameters implies elliott s identity and which also leads to concavity convexity properties of certain related functions we also show that elliott s identity is equivalent to a formula of ramanujan on the differentiation of quotients of hypergeometric functions applying these results we obtain a number of identities associated with the legendre functions of the first and the second kinds respectively

monotonicity criterion for the quotient of power series with applications abstract in this paper we present the necessary and sufficient condition for the monotonicity of the quotient of power series as applications some gaps and misquotations in certain published articles are pointed out and corrected and some known results involving the landen inequalities for zero balanced hypergeometric functions are improved

unification of the cubic analogues of the jacobian theta function abstract a common generalization a q z of hirschhorn garvan bonvein cubic analogues a q z b q z and c q z of the classical theta functions 2 q z 3 q z and 4 q z is introduced some properties of a q z are simply established and are shown to unify several known properties of a q z b q z and c q z

generalized convexity and inequalities abstract let r 0 and let m be the family of all mean values of two numbers in r some examples are the arithmetic geometric and harmonic means given m 1 m 2 m we say that a function f r r is m 1 m 2 convex if f m 1 x y m 2 f x f y for all x y r the usual convexity is the special case when both mean values are arithmetic means we study the dependence of m 1 m 2 convexity on m 1 and m 2 and give sufficient conditions for m 1 m 2 convexity of functions defined by maclaurin series the criteria involve the maclaurin coefficients our results yield a class of new inequalities for several special functions such as the gaussian hypergeometric function and a generalized bessel function

positive solutions and eigenvalue regions of two delay singular systems with a twin parameter abstract the paper is concerned with a two delay singular differential system with a twin parameter applying fixed point index theory we show the relationship between the asymptotic behaviors of nonlinearities at zero and infinity and the open regions eigenvalue regions of parameters which are correlated with delays such that the system has zero one and two positive solution s

a cubic counterpart of jacobi s identity and the agm we produce exact cubic analogues of jacobi s celebrated theta function identity and of the arithmetic geometric mean iteration of gauss and legendre the iteration in question is a n 1 a n 2b n 3 and b n 1 formula cannot be replicated the limit of this iteration is identified in terms of the hypergeometric function f 1 3 2 3 1 which supports a particularly simple cubic transformation

inclusion theorems of convolution operators associated with normalized hypergeometric functions for the normalized gaussian hypergeometric function zf a b c z given by f a b c z n 0 a n b n c n l n zn z 0 r z a convolution result for the class p is also given analogous results for the confluent hypergeometric functions are also discussed

hypergeometric transformation formulas of degrees 3 7 11 and 23 abstract the theory of theta functions is used to derive hypergeometric transformation formulas of degrees 3 7 11 and 23 as a consequence of the theory that is developed some new series for 1 are obtained that are similar to a class investigated by ramanujan

precise estimates for differences of the gaussian hypergeometric function abstract by showing certain combinations of the gaussian hypergeometric functions f a b a b 1 x c and f a b a b 1 x d to be monotone on 0 1 for given a b c d 0 a b and c d the authors study the problem of comparing these two functions they find sup 0 a f a b a b 1 x c f a b a b 1 x d for x 0 1 thus solving an open problem and a recent conjecture

critical groups and multiple solutions of the p laplacian equations abstract in this paper we give some results on the critical groups of the trivial solution u 0 of the asymmetric p laplacian equation p u u p 2 u u p 2 u with dirichlet boundary value some applications to the existence of multiple solutions of the equation p u f x u are given

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

some remarks on the fucik spectrum of the p laplacian and critical groups let be a bounded domain in n n 1 for 1 p the fucik spectrum of the p laplacian on w 1 p 0 is defined as the set p of those points a b 2 such that pu a u p 1 b u p 1 u w 1 p 0 1 1 has a nontrivial solution here pu div u p 2 u and u max u 0 it is known that the first eigenvalue 1 of p on w 1 p 0 is positive simple and admits a positive eigenfunction 1 w 1 p 0 c1 see lindqvist 8 so p contains the two lines 1 and 1 a first nontrivial curve c2 in p through 2 2 where 2 is the second eigenvalue of p was recently constructed and variationally characterized by a

positive radial symmetric solutions to an exterior elliptic robin boundary value problem and application abstract we study the existence uniqueness and behavior of positive radial symmetric solutions of the exterior boundary value problem u h x f u in r n b 0 r 0 u n a u on and u x 0 as x when a 0 the robin boundary condition has a wrong sign and this makes the analysis of the problem interesting and non trivial we determine a sharp condition on the coefficient a for the existence of positive spherically symmetric solutions of the boundary value problem the important model case f u u 1 7 a 1 2 r 0 and 1 2 r 0 is related to the initial data problem in general relativity for this model case as an example of application of our results we obtain the existence and uniqueness of the conformal factor corresponding to the hamiltonian constraint equation in the case of a single black hole in the radial symmetric regime