generalized lame functions i the elliptic case we present and study a class of functions associated with the two particle quantum relativistic calogero moser system with elliptic interactions the functions may be viewed as joint eigenfunctions of two independent commuting analytic difference operators one of which is the defining quantum dynamics the second one is obtained by interchanging the step size and the imaginary period the functions depend on parameters that are dense in the natural parameter domain in essence they consist of products of weierstrass functions and plane waves the zeros of the functions satisfy a constraint system encoding both schrodinger equations at once

a relativistic hypergeometric function we survey our work on a function generalizing 2f1 this function is a joint eigenfunction of four askey wilson type hyperbolic difference operators reducing to the askey wilson polynomials for certain discrete values of the variables it is defined by a contour integral generalizing the barnes representation of 2f1 it has various symmetries including a hidden d4 symmetry in the parameters by means of the associated hilbert space transform the difference operators can be promoted to self adjoint operators provided the parameters vary over a certain polytope in the parameter space for a dense subset of parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials hyperbolic functions and plane waves

quantization of models with non compact quantum group symmetry modular xxz magnet and lattice sinh gordon model we define and study certain integrable lattice models with non compact quantum group symmetry the modular double of including an integrable lattice regularization of the sinh gordon model and a non compact version of the xxz model their fundamental r matrices are constructed in terms of the non compact quantum dilogarithm our choice of the quantum group representations naturally ensures self adjointness of the hamiltonian and the higher integrals of motion these models are studied with the help of the separation of variables method we show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations baxter equation and quantum wronskian equation which is characterized by suitable analytic and asymptotic properties a key technical tool is the so called operator for which we give an explicit construction our results allow us to establish some connections to related results and conjectures on the sinh gordon theory in continuous spacetime our approach also sheds some light on the relations between massive and massless models in particular the sinh gordon and liouville theories from the point of view of their integrable structures

second quantization of the elliptic calogero sutherland model we construct a quantum field theory model of anyons on a circle and at finite temperature we find an anyon hamiltonian providing a second quantization of the elliptic calogero sutherland model this allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic calogero sutherland hamiltonian

an elliptic analogue of the multiple gamma function a hierarchy of functions including the elliptic gamma function is introduced it can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q analogue of the multiple gamma function some properties of the functions are considered

relativistic lame functions the special case g 2 we study a class of eigenfunctions of an analytic difference operator generalizing the special lam operator paying particular attention to quantum mechanical aspects we show that in a suitable scaling limit the pertinent eigenfunctions lead to the eigenfunctions of the operator in a finite volume we establish various orthogonality and non orthogonality results by direct calculations generalize the one gap picture associated with the above lam operator and obtain duality properties for the hyperbolic trigonometric and rational specializations

cas picard method for fractional nonlinear differential equation in this paper a computational method for solving the fractional nonlinear differential equation is introduced we proposed a method by utilizing the cas wavelets in conjunction with picard technique we call the method as cas picard method the fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by picard technique and then transformed into a system of algebraic equations by using the operational matrices of cas wavelets the error and supporting analysis of the method are also investigated the comparison analysis of method with other existing numerical methods is also performed

a new approach to the theory of functional integral equations of fractional order abstract the aim of the paper is to present a new approach to the theory of functional integral equations of fractional order that approach depends on converting of the mentioned equations to the form of functional integral equations of volterra stieltjes type it turns out that the study of functional integral equations of volterra stieltjes type is more convenient and effective than the study of functional integral equations of fractional order an example illustrating our approach is also discussed

essential self adjointness of the elliptic ruijsenaars models we study the elliptic ruijsenaars models associated with arbitrary root systems which are difference analogs of the calogero moser model we give a dense subspace in the space of square integrable functions invariant under the action of the weyl group on a torus as a domain of its hamiltonian and prove its essential self adjointness by using perturbation theory it is also clarified that these models consist of pure point spectrum

elliptic hypergeometry of supersymmetric dualities ii orthogonal groups knots and vortices we consider seiberg electric magnetic dualities for 4d mathcal n 1 sym theories with so n gauge group for all such known theories we construct superconformal indices scis in terms of elliptic hypergeometric integrals equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals in particular we describe a number of new elliptic beta integrals associated with the s confining theories with the spinor matter fields reductions of some dualities from sp 2n to so 2n or so 2n 1 gauge groups are described interrelation of scis and the witten anomaly is briefly discussed possible applications of the elliptic hypergeometric integrals to a two parameter deformation of 2d conformal field theory and related matrix models are indicated connections of the reduced scis with the state integrals of knot theory generalized agt duality for 3 3 d theories and a 2d vortex partition function are described

ruijsenaars commuting difference operators and invariant subspace spanned by theta functions we study a family of mutually commutative difference operators introduced by ruijsenaars the conjugations of these operators with an appropriate function give the hamiltonians of some relativistic quantum systems these operators can be regarded as elliptic analogs of the macdonald operators and their coefficients consist of the jacobi theta functions we show that these operators act on the space of meromorphic functions on the cartan subalgebra of affine lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators

holomorphic curves with shift invariant hyperplane preimages if f c p n is a holomorphic curve of hyper order less than one for which 2n 1 hyperplanes in general position have forward invariant preimages with respect to the translation z z c then f is periodic with period c 2 c this result which can be described as a difference analogue of m green s picard type theorem for holomorphic curves follows from a more general result presented in this paper the proof relies on a new version of cartan s second main theorem for the casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives both of which are proved here finally an application to the uniqueness theory of meromorphic functions is given and the sharpness of the obtained results is demonstrated by examples

elliptic hypergeometric series on root systems abstract we derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems a n c n and d n in the special cases of classical and q series our approach leads to new elementary proofs of the corresponding identities

relativistic lame functions revisited a basis for the eigenfunctions of the relativistic elliptic two particle calogero moser hamiltonian is known for a dense set in the natural parameter space we study the question whether an interpolating basis exists employing an asymptotic power series ansatz for the hyperbolic specialization we determine all of the coefficients explicitly which gives rise to formal interpolating eigenfunctions for the elliptic case we also need a power series ansatz for the eigenvalues we obtain the first few coefficients explicitly thus obtaining evidence for the existence of interpolating formal eigenfunctions and eigenvalues

time sliced path integrals with stationary states the path integral approach to the quantization of one degree of freedom newtonian particles is considered within the discrete time slicing approach as in feynman s original development in the time slicing approximation the quantum mechanical evolution will generally not have any stationary states we look for conditions on the potential energy term such that the quantum mechanical evolution may possess stationary states without having to perform a continuum limit when the stationary states are postulated to be solutions of a second order ordinary differential equation ode eigenvalue problem it is found that the potential is required to be a solution of a particular first order ode similarly when the stationary states are postulated to be solutions of a second order ordinary difference equation o e eigenvalue problem the potential is required to be a solution of a particular first order o e the classical limits which are at times very nontrivial are integrable maps

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

shintani barnes zeta and gamma functions abstract we show that shintani s work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations we re prove many of shintani s formulas and prove several new ones among the latter is a generalization to the shintani barnes gamma functions of raabe s 1843 formula 0 1 log x dx log 2 and a further generalization to the shintani zeta functions these explicit formulas can be interpreted as vanishing period integral side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions we also relate barnes triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems

traces on the sklyanin algebra and correlation functions of the eight vertex model we propose a conjectural formula for correlation functions of the z invariant inhomogeneous eight vertex model we refer to this conjecture as ansatz it states that correlation functions are linear combinations of products of three transcendental functions with theta functions and derivatives as coefficients the transcendental functions are essentially logarithmic derivatives of the partition function per site the coefficients are given in terms of a linear functional tr on the sklyanin algebra which interpolates the usual trace on finite dimensional representations we establish the existence of tr and discuss the connection to the geometry of the classical limit we also conjecture that the ansatz satisfies the reduced qkz equation as a non trivial example of the ansatz we present a new formula for the next nearest neighbour correlation functions

unitary representations of uq 2 ℝ the modular double and the multiparticle q deformed toda chain the paper deals with the analytic theory of the quantum q deformed toda chains the technique used combines the methods of representation theory and the quantum inverse scattering method the key phenomenon which is under scrutiny is the role of the modular duality concept first discovered by l faddeev in the representation theory of noncompact semisimple quantum groups explicit formulae for the whittaker vectors are presented in terms of the double sine functions and the wave functions of the n particle q deformed open toda chain are given as a multiple integral of the mellin barnes type for the periodic chain the two dual baxter equations are derived

parameter shifts d4 symmetry and joint eigenfunctions for commuting askey wilson type difference operators in previous papers we studied a generalized hypergeometric function that is a joint eigenfunction of four hyperbolic askey wilson difference operators using parameter shifts related to the d4 weight lattice we show that this function is the even linear combination of two elementary joint eigenfunctions for a dense subset of the parameter space more specifically the latter eigenfunctions are equal to a plane wave multiplied by hyperbolic functions

a generalized hypergeometric function satisfying four analytic difference equations of askey wilson type the hypergeometric function 2 f 1 can be written in terms of a contour integral involving gamma functions we generalize this barnes representation by using a certain generalized gamma function as a building block in this way we obtain a new 2 f 1 generalization with various symmetry features we determine the analyticity properties of the r function in all of its eight arguments and show that it is a joint eigenfunction of four distinct askey wilson type difference operators two acting on v and two on the askey wilson polynomials can be obtained by a suitable discretization of v or

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

kernel functions and backlund transformations for relativistic calogero moser and toda systems we obtain kernel functions associated with the quantum relativistic toda systems both for the periodic version and for the nonperiodic version with its dual this involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic calogero moser systems we show that the special kernel functions at issue admit a limit that yields generating functions of backlund transformations for the classical relativistic calogero moser and toda systems we also obtain the nonrelativistic counterparts of our results which tie in with previous results in the literature

numerical solution of integro differential equations by using cas wavelet operational matrix of integration the cas wavelet operational matrix p of integration is first presented and a general procedure to generate this matrix p is given cas wavelet approximating method is then utilized to reduce the integro differential equation to the algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

a new solution of the star triangle relation we obtain a new solution to the star triangle relation for an ising type model with two kinds of spin variables at each lattice site taking continuous real values and arbitrary integer values respectively the boltzmann weights are manifestly real and positive they are expressed through the euler gamma function and depend on sums and differences of spins at the ends of an edge of the lattice

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

direct method to construct integrals for nth order autonomous ordinary difference equations a direct method to construct integrals for an nth order autonomous ordinary difference equation oe wnnf wn wnn1 is presented as an illustration we first consider third order autonomous oe wn3

transformations of elliptic hypergeometric integrals we prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions corresponding to the root systems bc n and a n as a special case we recover some integral identities conjectured by van diejen and spiridonov for bc n we also consider their type ii integral their proof of that integral together with our transformation gives rise to pairs of adjoint integral operators a different proof gives rise to pairs of adjoint difference operators these allow us to construct a family of biorthogonal abelian functions generalizing the koornwinder polynomials and satisfying the analogues of the macdonald conjectures finally we discuss some transformations of type ii style integrals in particular we find that adding two parameters to the type ii integral gives an integral invariant under an appropriate action of the weyl group e 7

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

q deformed kzb heat equation completeness modular properties and sl 3 z abstract we study the properties of one dimensional hypergeometric integral solutions of the q difference quantum analogue of the knizhnik zamolodchikov bernard equations on tori we show that they also obey a difference kzb heat equation in the modular parameter give formulae for modular transformations and prove a completeness result by showing that the associated fourier transform is invertible these results are based on sl 3 z transformation properties parallel to those of elliptic gamma functions

the perturbation of the quantum calogero moser sutherland system and related results the hamiltonian of the trigonometric calogero sutherland model coincides with a certain limit of the hamiltonian of the elliptic calogero moser model in other words the elliptic hamiltonian is a perturbed operator of the trigonometric one

krull s theory for the double gamma function abstract the barnes g function g x 1 2 satisfies the functional equation log g x 1 log g x log x we complement w krull s work in bemerkungen zur differenzengleichung g x 1 g x x math nachrichten 1 1948 365 376 with additional results that yield a different characterization of the function g new expansions and sharp bounds for g on x 0 in terms of gamma and digamma functions a new expansion for the gamma function and summation formulae with polygamma functions

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

exact solution of the quantum calogero gaudin system and of its q deformation a complete set of commuting observables for the calogero gaudin system is diagonalized and the explicit form of the corresponding eigenvalues and eigenfunctions is derived we use a purely algebraic procedure exploiting the coalgebra invariance of the model with the proper technical modifications this procedure can be applied to the q deformed version of the model which is then also exactly solved

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

new elliptic solutions of the yang baxter equation we consider finite dimensional reductions of an integral operator with the elliptic hypergeometric kernel describing the most general known solution of the yang baxter equation with a rank 1 symmetry algebra the reduced r operators reproduce at their bottom the standard baxter s r matrix for the 8 vertex model and sklyanin s l operator the general formula has a remarkably compact form and yields new elliptic solutions of the yang baxter equation based on the finite dimensional representations of the elliptic modular double the same result is also derived using the fusion formalism

new solutions of the star triangle relation with discrete and continuous spin variables a new solution to the star triangle relation is given for an ising type model of interacting spins containing integer and real valued components boltzmann weights of the model are given in terms of the lens elliptic gamma function and are based on yamazaki s recently obtained solution of the star star relation the star triangle relation given here implies seiberg duality for the 4 d index of the su 2 quiver gauge theory and the corresponding two component spin case of the star star relation of yamazaki a proof of the star triangle relation is given resulting in a new elliptic hypergeometric summation integration identity the star triangle relation in this paper contains the master solution of bazhanov and sergeev as a special case two other limiting cases are considered one of which gives a new star triangle relation in terms of ratios of infinite q products while the other case gives a new way of deriving a star triangle relation that was previously obtained by the author

generalized lame functions ii hyperbolic and trigonometric specializations in part i j math phys 40 1595 1999 we studied eigenfunctions of the quantum dynamics that defines the two particle relativistic calogero moser system with elliptic interaction in the present paper we consider the same system with hyperbolic and trigonometric interactions in these special regimes the eigenfunctions are shown to admit an elementary representation that is far more explicit than the zero representation of part i in particular the new representation can be exploited to prove that the hyperbolic eigenfunctions can be chosen to be symmetric under interchanging position and momentum variables self duality in the trigonometric case duality properties are derived too and several orthogonality and completeness results are obtained

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

numerical solution of fredholm integral equations by using cas wavelets a numerical method for solving the fredholm integral equations is presented the method is based upon cas wavelet approximations the properties of cas wavelet are first presented cas wavelet approximations method are then utilized to reduce the fredholm integral equations to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

inversions of integral operators and elliptic beta integrals on root systems we prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems using the a c inversion formula to invert one of the known c type elliptic beta integrals we obtain a new elliptic beta integral for the root system of type a validity of this integral is established by a different method as well

on razamat s a 2 and a 3 kernel identities in recent work on superconformal quantum field theories razamat arrived at elliptic kernel identities of a new and striking character they relate solely to the root systems a 2 and a 3 and have no coupling type parameters cite ra18 the pertinent 2 and 3 variable hamiltonians are analytic difference operators and the kernel functions are built from the elliptic gamma function razamat presented compelling evidence for the validity of these identities and checked them to a certain order in a power series expansion this paper is mainly concerned with analytical proofs of these identities more specifically we furnish a complete proof of the identities for the a 2 elliptic case and for the a 3 hyperbolic case and consider several specializations we also discuss the implications the kernel identities might have for a hilbert space scenario involving common eigenvectors of the hamiltonians and the integral operators associated with the kernel functions

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

q special functions with q 1 and their application to discrete integrable systems solutions for certain discrete integrable systems are constructed by using integral solutions for hypergeometric q difference systems with q 1 we apply a solution for lauricella s d type hypergeometric q difference system with q 1 to construct a solution for the discrete kp hierarchy furthermore an integral solution of a q difference analogue of bessel s equation is newly introduced and applied to construct a solution for a q difference analogue of the cylindrical toda equation with q 1

ruijsenaars commuting difference operators as commuting transfer matrices for belavin s elliptic quantum r matrix we construct an l operator as a set of difference operators acting on functions on the type a weight space according to the fundamental relation rll llr taking the trace of the l operator gives a set of commuting difference operators we show that for the above mentioned l operator this approach gives macdonald type operators with elliptic theta function coefficient actually equivalent to ruijsenaars operators the relationship between the difference l operator and krichever s lax matrix is given and an explicit formula for elliptic commuting differential operators is derived we also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space

the technique of volterra stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders our study in this paper concerns the infinite systems of nonlinear integral equations of volterra stieltjes type we show that some assumptions imposed on terms of the mentioned systems guarantee the solvability of those systems in the space c i c 0 with i 0 1 and c 0 being the classical sequence space as a particular and very important case of the investigated systems of nonlinear volterra stieltjes integral equations we obtain infinite systems of nonlinear integral equations of fractional orders thus the approach applied in the paper enables us to obtain rather general results concerning the existence of solutions of infinite systems of nonlinear fractional integral equations

solvability of volterra stieltjes operator integral equations and their applications abstract we investigate a class of operator integral equations of volterra stieltjes type and we study the solvability of those equations in the space of continuous functions equations in question create a generalization of numerous integral equations considered in nonlinear analysis the main tool used in our considerations is the technique associated with measures of noncompactness we show the applicability of our existence result in the study of a few integral equations of volterra type

an addition type formula for the elliptic digamma function abstract eisenstein derived addition formulas for the weierstrass zeta function from the addition formula for the cotangent function and the fact that the weierstrass zeta function can be represented as an infinite sum of the cotangent functions in this paper we apply this idea of eisenstein s to the addition type formula for the double cotangent function established by the author we show that the elliptic digamma function defined by the logarithmic derivative of the elliptic gamma function satisfies an addition type formula this formula includes the addition formula for the weierstrass zeta function evaluation formulas for the double eisenstein series introduced by tsumura and the double shuffle relations for the double eisenstein series proved by gangl kaneko zagier

some properties of nonlinear volterra stieltjes integral operators we study some properties of the nonlinear volterra stieltjes integral operator of general form which is defined with help of the stieltjes integral with the kernel depending on two variables the results obtained in the paper generalize those obtained earlier for the volterra stieltjes integral operators having less general form we prove also an existence result for nonlinear integral equation of volterra stieltjes type