2016 
Zudilin W, 'A Determinantal Approach to Irrationality', Constructive Approximation, 110 (2016)
Â© 2016 Springer Science+Business Media New YorkIt is a classical fact that the irrationality of a number (Formula presented.) follows from the existence of a sequence (Formula pr... [more]
Â© 2016 Springer Science+Business Media New YorkIt is a classical fact that the irrationality of a number (Formula presented.) follows from the existence of a sequence (Formula presented.) with integral (Formula presented.) and (Formula presented.) such that (Formula presented.) for all n and (Formula presented.) as (Formula presented.). In this paper, we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement (Formula presented.) is weakened. Some applications are given, including a new proof of the irrationality of (Formula presented.). Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.



2016 
Brent RP, Coons M, Zudilin W, 'Algebraic Independence of Mahler Functions via Radial Asymptotics', International Mathematics Research Notices, 2016 571603 (2016) [C1]
Â© 2015 The Author(s). Published by Oxford University Press. All rights reserved.We present a new method for algebraic independence results in the context of Mahler's method. In p... [more]
Â© 2015 The Author(s). Published by Oxford University Press. All rights reserved.We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behavior of a Mahler function f (z) as z goes radially to a root of unity to deduce algebraic independence results about the values of f (z) at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to F (z), the power series solution to the functional equation F (z)  (1 + z+ z2)F (z4) + z4F (z16)=0. Specifically, we prove that the functions F(z), F (z4), F' (z), and F'(z4) are algebraically independent over C(z). An application of a celebrated result of Ku. Nishioka then allows one to replace C(z) by Q when evaluating these functions at a nonzero algebraic number a in the unit disc.



2016 
LalÃn M, Samart D, Zudilin W, 'Further explorations of Boyd's conjectures and a conductor 21 elliptic curve', Journal of the London Mathematical Society, 93 341360 (2016)
Â© 2016 London Mathematical Society.We prove that the (logarithmic) Mahler measure {rm m}(P) of P(x,y)=x+1/x+y+1/y+3 is equal to the Lvalue 2L'(E,0) attached to the elliptic curv... [more]
Â© 2016 London Mathematical Society.We prove that the (logarithmic) Mahler measure {rm m}(P) of P(x,y)=x+1/x+y+1/y+3 is equal to the Lvalue 2L'(E,0) attached to the elliptic curve E:P(x,y)=0 of conductor 21. In order to do this, we investigate the measure of a more general Laurent polynomial [Pa,b,c}(x,y)=aleft(x+\frac1xright)+bleft(y+frac1yright)+c] and show that the wanted quantity {m}(P) is related to a 'halfMahler' measure of P(x,y)=P 7,1,3}(x,y). In the finale, we use the modular parametrization of the elliptic curve P(x,y)=0, again of conductor 21, due to Ramanujan and the MellitBrunault formula for the regulator of modular units.



2016 
Bertin MJ, Zudilin W, 'On the Mahler measure of a family of genus 2 curves', Mathematische Zeitschrift, 19 (2016)



2016 
Osburn R, Zudilin W, 'On the (K.2) supercongruence of Van Hamme', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 433 706711 (2016)



2015 
Haynes A, Zudilin W, 'Hankel determinants of zeta values', Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 11 15 (2015) [C1]



2015 
Zudilin W, 'On three theorems of Folsom, Ono and Rhoades', Proceedings of the American Mathematical Society, 143 14711476 (2015) [C1]
Â© 2014 American Mathematical Society.In his deathbed letter to G.H. Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotic matc... [more]
Â© 2014 American Mathematical Society.In his deathbed letter to G.H. Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotic matches the one of a modular form, though a choice of the modular function depends on the root of unity. Recently Folsom, Ono and Rhoades have proved an elegant result about the match for a general family related to DysonÂ¿s rank (mock theta) function and the AndrewsÂ¿Garvan crank (modular) functionÂ¿ the match with explicit formulae for implied O(1) constants. In this note we give another elementary proof of RamanujanÂ¿s original claim and outline some heuristics which may be useful for obtaining a new proof of the general FolsomÂ¿OnoÂ¿Rhoades theorem.



2015 
Viola C, Zudilin W, 'Linear independence of dilogarithmic values', Journal fÃ¼r die reine und angewandte Mathematik (Crelles Journal), 0 (2015)



2015 
Zudilin W, 'Multiple qZeta Brackets', Mathematics, 3 119130 (2015) [C1]



2015 
Straub A, Zudilin W, 'Positivity of rational functions and their diagonals', Journal of Approximation Theory, 195 5769 (2015) [C1]
Â© 2014.The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szego as... [more]
Â© 2014.The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szego as well as Askey and Gasper, who inspired more recent work. It is well known that the diagonal coefficients of rational functions are Dfinite. This note is motivated by the observation that, for several of the rational functions whose positivity has received special attention, the diagonal terms in fact have arithmetic significance and arise from differential equations that have modular parametrization. In each of these cases, this allows us to conclude that the diagonal is positive.Further inspired by a result of Gillis, Reznick and Zeilberger, we investigate the relation between positivity of a rational function and the positivity of its diagonal.



2014 
Zudilin W, 'A GENERATING FUNCTION OF THE SQUARES OF LEGENDRE POLYNOMIALS', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 89 125131 (2014) [C1]



2014 
Dauguet S, Zudilin W, 'On simultaneous diophantine approximations to Â¿(2) and Â¿(3)', Journal of Number Theory, 145 362387 (2014) [C1]
We present a hypergeometric construction of rational approximations to Â¿(2) and Â¿(3) which allows one to demonstrate simultaneously the irrationality of each of the zeta values,... [more]
We present a hypergeometric construction of rational approximations to Â¿(2) and Â¿(3) which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, Â¿(2) and Â¿(3) with rational coefficients. We then go further to formalize the arithmetic structure of these specific linear forms by introducing a new notion of (simultaneous) diophantine exponent. Finally, we study the properties of this newer concept and link it to the classical irrationality exponent and its generalizations given recently by S. Fischler. Â© 2014 Elsevier Inc.



2014 
Zudilin W, 'Regulator of modular units and Mahler measures', Mathematical Proceedings of the Cambridge Philosophical Society, 156 313326 (2014) [C1]



2014 
Rogers M, Zudilin W, 'On the Mahler Measure of 1+X+1/X+Y +1/Y', International Mathematics Research Notices, 2014 23052326 (2014) [C1]



2014 
Zudilin W, 'Two hypergeometric tales and a new irrationality measure of Â¿(2)', Annales mathÃ©matiques du QuÃ©bec, 38 101117 (2014) [C1]



2013 
Chan HH, Wan J, Zudilin W, 'Legendre polynomials and Ramanujantype series for 1/p', Israel Journal of Mathematics, 194 183207 (2013) [C1]



2013 
Wan J, Zudilin W, 'Generating functions of Legendre polynomials: A tribute to Fred Brafman', Journal of Approximation Theory, 170 198213 (2013) [C1]



2013 
Zudilin V, 'On the irrationality measure of p^2', Russian Mathematical Surveys, 68 11331135 (2013) [C1]



2012 
Guillera J, Zudilin V, ''Divergent' Ramanujantype supercongruences', Proceedings of the American Mathematical Society, 140 765777 (2012) [C1]



2012 
Borwein JM, Straub A, Wan G, Zudilin V, 'Densities of short uniform random walks', Canadian Journal of Mathematics, 64 961990 (2012) [C1]



2012 
Rogers M, Zudilin V, 'From Lseries of elliptic curves to Mahler measures', Compositio Mathematica, 148 385414 (2012) [C1]



2012 
Wan G, Zudilin V, 'Generating functions of Legendre polynomials: A tribute to Fred Brafman', Journal of Approximation Theory, 164 488503 (2012) [C1]



2012 
Ohno Y, Okuda JI, Zudilin V, 'Cyclic qMZSV sum', Journal of Number Theory, 132 144155 (2012) [C1]



2012 
Warnaar SO, Zudilin V, 'Dedekind's Â¿function and RogersRamanujan identities', Bulletin of the London Mathematical Society, 44 111 (2012) [C1]



2012 
Chan HH, Wan G, Zudilin V, 'Complex series for 1/p', Ramanujan Journal, 29 135144 (2012) [C1]



2011 
Chan HH, Tanigawa Y, Yang Y, Zudilin V, 'New analogues of Clausen's identities arising from the theory of modular forms', Advances in Mathematics, 228 12941314 (2011) [C1]



2011 
Warnaar SO, Zudilin V, 'A qrious positivity', Aequationes Mathematicae, 81 177183 (2011) [C1]



2011 
Almkvist G, Van Straten D, Zudilin V, 'Generalizations of Clausen's Formula and algebraic transformations of CalabiYau differential equations', Proceedings of the Edinburgh Mathematical Society, 54 273295 (2011) [C1]



2011 
Zudilin V, 'Book Review: Ramanujan's Lost Notebook. Part II, G.E. Andrews, B.C. Berndt', Journal of Approximation Theory, 163 10371038 (2011) [C3] 


2011 
Gallot Y, Moree P, Zudilin V, 'The Erd'sMoser equation 1k +2k +...+(m1)k = mk revisited using continued fractions', Mathematics of Computation, 80 12211237 (2011) [C1]



2011 
Zudilin V, 'Arithmetic hypergeometric series', Russian Mathematical Surveys, 66 369420 (2011) [C1]



2010 
Chan HH, Long L, Zudilin V, 'A supercongruence motivated by the Legendre family of elliptic curves', Mathematical Notes, 88 599602 (2010) [C1]



2010 
Chan HH, Zudilin V, 'New representations for aperylike sequences', Mathematika, 56 107117 (2010) [C1]



2010 
Fischler S, Zudilin V, 'A refinement of Nesterenko's linear independence criterion with applications to zeta values', Mathematische Annalen, 347 739763 (2010) [C1]



2010 
Bailey DH, Borwein JM, Broadhurst D, Zudilin V, 'Experimental mathematics and mathematical physics', Contemporary Mathematics, 517 4158 (2010) [C1]



2010 
Yang Y, Zudilin V, 'On Sp4 modularity of PicardFuchs differential equations for CalabiYan threefolds', Contemporary Mathematics, 517 381413 (2010) [C1]



2009 
Zudilin V, 'Apery's theorem. Thirty years after', International Journal of Mathematics and Computer Science, 4 919 (2009) [C1] 


2009 
Zudilin V, 'Ramanujantype supercongruences', Journal of Number Theory, 129 18481857 (2009) [C1]



2009 
Krattenthaler C, Rochev I, Vaananen K, Zudilin V, 'On the nonquadraticity of values of the qexponential function and related qseries', Acta Arithmetica, 136 243269 (2009) [C1]



2009 
Zudilin V, 'A hypergeometric problem', Journal of Computational and Applied Mathematics, 233 856857 (2009) [C1]



2008 
Ohno Y, Zudilin W, 'Zeta Stars', Communications in Number Theory and Physics, 2 325347 (2008) [C1]



2008 
Zudilin W, 'Linear independence of values of Tschakaloff functions with different parameters', Journal of Number Theory, 128 25492558 (2008) [C1]



2008 
Viola C, Zudilin W, 'Hypergeometric transformations of linear forms in one logarithm', Functiones et Approximatio Commentarii Mathematici, 39 211222 (2008) [C1] 


2007 
Zudilin W, 'Approximations to , di and trilogarithms', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 202 450459 (2007) [C1]



2007 
Zudilin W, 'An elementary proof of the irrationality of Tschakaloff series', Journal of Mathematical Sciences, 146 56695673 (2007) [C1]



2007 
Zudilin W, 'A new lower bound for II(3/2)kII', Journal de Theorie des Nombres de Bordeaux, 19 313325 (2007) [C1] 


2007 
Zudilin VV, 'Quadratic transformations and Guillera's formulas for 1/pi(2)', MATHEMATICAL NOTES, 81 297301 (2007) [C1]



2007 
Bundschuh P, Zudilin W, 'Irrationality measures for certain qmathematical constants', MATHEMATICA SCANDINAVICA, 101 104122 (2007) [C1]



2007 
Vaananen K, Zudilin WV, 'Linear independence of values of Tschakaloff series', RUSSIAN MATHEMATICAL SURVEYS, 62 196198 (2007) [C1]



2007 
Zudilin VV, 'More Ramanujantype formulae for 1/p 2', Russian Mathematical Surveys, 62 634636 (2007) [C1]



2006 
MatalaAho T, Vaananen K, Zudilin W, 'New irrationality measures for qlogarithms', MATHEMATICS OF COMPUTATION, 75 879889 (2006) [C1]



2006 
Pilehrood KH, Pilehrood TH, Zudilin W, 'Irrationality of certain numbers that contain values of the di and trilogarithm', MATHEMATISCHE ZEITSCHRIFT, 254 299313 (2006) [C1]



2006 
Sondow J, Zudilin W, 'Euler's constant, qlogarithms, and formulas of Ramanujan and Gosper', RAMANUJAN JOURNAL, 12 225244 (2006) [C1]



2006 
Krattenthaler C, Rivoal T, Zudilin W, 'Series hypergeometriques basiques, qanalogues des valeurs de la fonction zeta et formes modulaires', Journal of the Institute of Mathematics of Jussieu, 5 5379 (2006) [C1] 


2006 
Zudilin W, 'Approximations to qlogarithms and qdilogarithms, with applications to qzeta values', Journal of Mathematical Sciences, 137 46734683 (2006) [C1]
We construct simultaneous rational approximations to qseries L 1(x1; q) and L1(x2; q) and, if x = x1 = x2, to series L1(x; q) and L 2(x; q), where L1 (x;q) = Â¿8n=1(xq) n/1qn=Â¿... [more]
We construct simultaneous rational approximations to qseries L 1(x1; q) and L1(x2; q) and, if x = x1 = x2, to series L1(x; q) and L 2(x; q), where L1 (x;q) = Â¿8n=1(xq) n/1qn=Â¿8n=1xqn/1xqn L2 (x;q) = Â¿8n=1n(xq)n/1qn= Â¿8n=1xqn/1xqn. Applying the construction, we obtain quantitative linear independence over Q of the numbers in the following collections: 1, Â¿q(1) = L1(1; q), Â¿ (q 2) and 1, Â¿q(1), Â¿q(2) = L 2(1; q) for q = 1/p, p e Z \ (0,Â±1). Bibliography: 14 titles. Â© 2006 Springer Science+Business Media, Inc.



2005 
Vaananen K, Zudilin W, 'Bakertype estimates for linear forms in the values of qseries', CANADIAN MATHEMATICAL BULLETINBULLETIN CANADIEN DE MATHEMATIQUES, 48 147160 (2005) [C1]



2005 
Zudilin W, 'Wellpoised generation of Aperylike recursions', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 178 513521 (2005) [C1]



2005 
Zudilin WV, 'Ramanujantype formulae and irrationality measures of some multiples of p', Sbornik Mathematics, 196 983998 (2005)



2005 
Zudilin W, 'Computing powers of two generalizations of the logarithm', Seminaire Lotharingien de Combinatoire, 53 16 (2005) [C1] 


2004 
Zudilin VV, 'Binomial sums related to rational approximations to Â¿(4)', Mathematical Notes, 75 594597 (2004)



2004 
Zudilin VV, 'The inverse legendre transform of a certain family of sequences', Mathematical Notes, 76 276279 (2004) [C1]



2004 
Zudilin W, 'Binomial sums related to rational approximations to Zeta(4)', Russian Academy of Sciences: Mathematical Notes, 75 594597 (2004) [C1]



2004 
Bundschuh P, Zudilin W, 'On theorems of Gelfond and Selberg concerning integralvalued entire functions', JOURNAL OF APPROXIMATION THEORY, 130 164178 (2004) [C1]



2004 
Zudilin W, 'Wellpoised hypergeometric transformations of Eulertype multiple integrals', JOURNAL OF THE LONDON MATHEMATICAL SOCIETYSECOND SERIES, 70 215230 (2004) [C1]



2004 
Zudilin W, 'Heine's basic transform and a permutation group for qharmonic series', ACTA ARITHMETICA, 111 153164 (2004) [C1]



2004 
Zudilin W, 'On a combinatorial problem of Asmus Schmidt', ELECTRONIC JOURNAL OF COMBINATORICS, 11 (2004) [C1]



2004 
Zudilin W, 'On a combinatorial problem of Asmus Schmidt', Electronic Journal of Combinatorics, 11 (2004)



2004 
Zudilin W, 'Arithmetic of linear forms involving odd zeta values', Journal de Theorie des Nombres de Bordeaux, 16 251291 (2004) [C1]



2003 
Zudilin W, 'The hypergeometric equation and Ramanujan functions', RAMANUJAN JOURNAL, 7 435447 (2003) [C1]



2003 
Zudilin VV, 'On the Functional Transcendence of qZeta Values', Mathematical Notes, 73 588589 (2003) [C1]



2003 
Rivoal T, Zudilin W, 'Diophantine properties of numbers related to Catalan's constant', MATHEMATISCHE ANNALEN, 326 705721 (2003) [C1]



2003 
Zudilin VV, 'Algebraic relations for multiple zeta values', Russian Mathematical Surveys, 58 129 (2003) [C1]



2003 
Bertrand D, Zudilin W, 'On the transcendence degree, of the differential field generated by Siegel modular forms', JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 554 4768 (2003) [C1]



2003 
Zudilin W, 'An Aperylike difference equation for Catalan's constant', ELECTRONIC JOURNAL OF COMBINATORICS, 10 (2003) [C1]



2003 
Zudilin W, 'An ApÃ©rylike difference equation for Catalan's constant', Electronic Journal of Combinatorics, 10 (2003)
Applying Zeilberger's algorithm of creative telescoping to a family of certain verywellpoised hypergeometric series involving linear forms in Catalan's constant with rational co... [more]
Applying Zeilberger's algorithm of creative telescoping to a family of certain verywellpoised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a secondorder difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan's constant as well as a new continuedfraction expansion for it. Similar arguments are put forward to deduce a secondorder difference equation and a new continued fraction for Â¿(4) = p4/90.



2003 
Zudilin W, 'Wellpoised hypergeometric service for diophantine problems of zeta values', Journal de Theorie des Nombres de Bordeaux, 15 593626 (2003) [C1]



2002 
Zudilin W, 'Remarks on irrationality of qharmonic series', MANUSCRIPTA MATHEMATICA, 107 463477 (2002)



2002 
Zudilin VV, 'Very wellpoised hypergeometric series and multiple integrals', Russian Mathematical Surveys, 57 824826 (2002)



2002 
Zudilin WV, 'On the irrationality measure for a qanalogue of Â¿(2)', Sbornik Mathematics, 193 11511172 (2002)



2002 
Zudilin W, 'Irrationality of values of the Riemann zeta function', Izvestiya Mathematics, 66 489542 (2002)
The paper deals with a generalization of Rivoal's construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function Â¿(s) only at od... [more]
The paper deals with a generalization of Rivoal's construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function Â¿(s) only at odd points. We prove theorems on the irrationality of the number Â¿(s) for some odd integers s in a given segment of the set of positive integers. Using certain refined arithmetical estimates, we strengthen Rivoal's original results on the linear independence of the Â¿(s). Â© 2002 RAS(DoM) and LMS.



2002 
Zudilin VV, 'Integrality of power expansions related to hypergeometric series', Mathematical Notes, 71 604616 (2002)
In the present paper, we study the arithmetic properties of power expansions related to generalized hypergeometric differential equations and series. Defining the series f(z), g(z... [more]
In the present paper, we study the arithmetic properties of power expansions related to generalized hypergeometric differential equations and series. Defining the series f(z), g(z) in powers of z so that f(z) and f(z) log z + g(z) satisfy a hypergeometric equation under a special choice of parameters, we prove that the series q(z) = ze g(Cz)/f(Cz) in powers of z and its inversion z(q) in powers of q have integer coefficients (here the constant C depends on the parameters of the hypergeometric equation). The existence of an integral expansion z(q) for differential equations of second and third order is a classical result; for orders higher than 3 some partial results were recently established by Lian and Yau. In our proof we generalize the scheme of their arguments by using Dwork's padic technique.



2002 
Zudilin VV, 'A thirdorder ApÃ©rylike recursion for Â¿(5)', Mathematical Notes, 72 733737 (2002)



2002 
Zudilin VV, 'Diophantine problems for qZeta values', Mathematical Notes, 72 858862 (2002)



2001 
Zudilin VV, 'One of the eight numbers Â¿(5), Â¿(7),Â¿ , Â¿(17), Â¿(19) is irrational', Mathematical Notes, 70 426431 (2001)



2001 
Bertrand D, Zudilin W, 'Derivatives of Siegel modular forms and exponential functions', Izvestiya Mathematics, 65 659671 (2001)
We show that the differential field generated by Siegel modular forms and the differential field generated by exponentials of polynomials are linearly disjoint over C. Combined wi... [more]
We show that the differential field generated by Siegel modular forms and the differential field generated by exponentials of polynomials are linearly disjoint over C. Combined with our previous work [3], this provides a complete multidimensional extension of Mahler's theorem on the transcendence degree of the field generated by the exponential function and the derivatives of a modular function. We give two proofs of our result, one purely algebraic, the other analytic, but both based on arguments from differential algebra and on the stability under the action of the symplectic group of the differential field generated by rational and modular functions. Â©2001 RAS(DoM) and LMS.



1997 
Zudilin VV, 'On the measure of linear and algebraic independence for values of entire hypergeometric functions', Mathematical Notes, 61 246248 (1997) 


1997 
Zudilin VV, 'Recurrent sequences and the measure of irrationality of values of elliptic integrals', Mathematical Notes, 61 657661 (1997) 


1996 
Zudilin VV, 'On the algebraic structure of functional matrices of special form', Mathematical Notes, 60 642648 (1996) 


1995 
Zudilin VV, 'On rational approximations of values of a certain class of entire functions', Sbornik Mathematics, 186 555590 (1995)


