2015 | Zudilin W, 'On three theorems of Folsom, Ono and Rhoades', Proceedings of the American Mathematical Society, 143 1471-1476 (2015)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 ... [more] 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. | | |
2014 | Zudilin W, 'A GENERATING FUNCTION OF THE SQUARES OF LEGENDRE POLYNOMIALS', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 89 125-131 (2014) [C1] | | |
2014 | Straub A, Zudilin W, 'Positivity of rational functions and their diagonals', Journal of Approximation Theory, (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{double acu... [more] 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{double acute} as well as Askey and Gasper, who inspired more recent work. It is well known that the diagonal coefficients of rational functions are D-finite. 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. Crown Copyright Â© 2014. | | |
2014 | Dauguet S, Zudilin W, 'On simultaneous diophantine approximations to Â¿(2) and Â¿(3)', Journal of Number Theory, 145 362-387 (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, 156 313-326 (2014) [C1] | | |
2014 | Rogers M, Zudilin W, 'On the Mahler Measure of 1+X+1/X+Y +1/Y', International Mathematics Research Notices, 2014 2305-2326 (2014) [C1] | | |
2014 | Zudilin W, 'Two hypergeometric tales and a new irrationality measure of Â¿(2)', Annales mathÃ©matiques du QuÃ©bec, 38 101-117 (2014) [C1] | | |
2013 | Chan HH, Wan J, Zudilin W, 'Legendre polynomials and Ramanujan-type series for 1/p', Israel Journal of Mathematics, 194 183-207 (2013) [C1] | | |
2013 | Wan J, Zudilin W, 'Generating functions of Legendre polynomials: A tribute to Fred Brafman', Journal of Approximation Theory, 170 198-213 (2013) [C1] | | |
2013 | Zudilin V, 'On the irrationality measure of p^2', Russian Mathematical Surveys, 68 1133-1135 (2013) [C1] | | |
2012 | Guillera J, Zudilin V, ''Divergent' Ramanujan-type supercongruences', Proceedings of the American Mathematical Society, 140 765-777 (2012) [C1] | | |
2012 | Borwein JM, Straub A, Wan G, Zudilin V, 'Densities of short uniform random walks', Canadian Journal of Mathematics, 64 961-990 (2012) [C1] | | |
2012 | Rogers M, Zudilin V, 'From L-series of elliptic curves to Mahler measures', Compositio Mathematica, 148 385-414 (2012) [C1] | | |
2012 | Wan G, Zudilin V, 'Generating functions of Legendre polynomials: A tribute to Fred Brafman', Journal of Approximation Theory, 164 488-503 (2012) [C1] | | |
2012 | Ohno Y, Okuda J-I, Zudilin V, 'Cyclic q-MZSV sum', Journal of Number Theory, 132 144-155 (2012) [C1] | | |
2012 | Warnaar SO, Zudilin V, 'Dedekind's Â¿-function and Rogers-Ramanujan identities', Bulletin of the London Mathematical Society, 44 1-11 (2012) [C1] | | |
2012 | Chan HH, Wan G, Zudilin V, 'Complex series for 1/p', Ramanujan Journal, 29 135-144 (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 1294-1314 (2011) [C1] | | |
2011 | Warnaar SO, Zudilin V, 'A q-rious positivity', Aequationes Mathematicae, 81 177-183 (2011) [C1] | | |
2011 | Almkvist G, Van Straten D, Zudilin V, 'Generalizations of Clausen's Formula and algebraic transformations of Calabi-Yau differential equations', Proceedings of the Edinburgh Mathematical Society, 54 273-295 (2011) [C1] | | |
2011 | Zudilin V, 'Book Review: Ramanujan's Lost Notebook. Part II, G.E. Andrews, B.C. Berndt', Journal of Approximation Theory, 163 1037-1038 (2011) [C3] | | |
2011 | Gallot Y, Moree P, Zudilin V, 'The Erd's-Moser equation 1k +2k +...+(m-1)k = mk revisited using continued fractions', Mathematics of Computation, 80 1221-1237 (2011) [C1] | | |
2011 | Zudilin V, 'Arithmetic hypergeometric series', Russian Mathematical Surveys, 66 369-420 (2011) [C1] | | |
2010 | Chan HH, Long L, Zudilin V, 'A supercongruence motivated by the Legendre family of elliptic curves', Mathematical Notes, 88 599-602 (2010) [C1] | | |
2010 | Chan HH, Zudilin V, 'New representations for apery-like sequences', Mathematika, 56 107-117 (2010) [C1] | | |
2010 | Fischler S, Zudilin V, 'A refinement of Nesterenko's linear independence criterion with applications to zeta values', Mathematische Annalen, 347 739-763 (2010) [C1] | | |
2010 | Bailey DH, Borwein JM, Broadhurst D, Zudilin V, 'Experimental mathematics and mathematical physics', Contemporary Mathematics, 517 41-58 (2010) [C1] | | |
2010 | Yang Y, Zudilin V, 'On Sp4 modularity of Picard-Fuchs differential equations for Calabi-Yan threefolds', Contemporary Mathematics, 517 381-413 (2010) [C1] | | |
2009 | Zudilin V, 'Apery's theorem. Thirty years after', International Journal of Mathematics and Computer Science, 4 9-19 (2009) [C1] | | |
2009 | Zudilin V, 'Ramanujan-type supercongruences', Journal of Number Theory, 129 1848-1857 (2009) [C1] | | |
2009 | Krattenthaler C, Rochev I, Vaananen K, Zudilin V, 'On the non-quadraticity of values of the q-exponential function and related q-series', Acta Arithmetica, 136 243-269 (2009) [C1] | | |
2009 | Zudilin V, 'A hypergeometric problem', Journal of Computational and Applied Mathematics, 233 856-857 (2009) [C1] | | |
2008 | Ohno Y, Zudilin W, 'Zeta Stars', Communications in Number Theory and Physics, 2 325-347 (2008) [C1] | | |
2008 | Zudilin W, 'Linear independence of values of Tschakaloff functions with different parameters', Journal of Number Theory, 128 2549-2558 (2008) [C1] | | |
2008 | Viola C, Zudilin W, 'Hypergeometric transformations of linear forms in one logarithm', Functiones et Approximatio Commentarii Mathematici, 39 211-222 (2008) [C1] | | |
2007 | Zudilin W, 'Approximations to -, di- and tri-logarithms', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 202 450-459 (2007) [C1] | | |
2007 | Zudilin W, 'An elementary proof of the irrationality of Tschakaloff series', Journal of Mathematical Sciences, 146 5669-5673 (2007) [C1] | | |
2007 | Zudilin W, 'A new lower bound for II(3/2)kII', Journal de Theorie des Nombres de Bordeaux, 19 313-325 (2007) [C1] | | |
2007 | Zudilin VV, 'Quadratic transformations and Guillera's formulas for 1/pi(2)', MATHEMATICAL NOTES, 81 297-301 (2007) [C1] | | |
2007 | Bundschuh P, Zudilin W, 'Irrationality measures for certain q-mathematical constants', MATHEMATICA SCANDINAVICA, 101 104-122 (2007) [C1] | | |
2007 | Vaananen K, Zudilin WV, 'Linear independence of values of Tschakaloff series', RUSSIAN MATHEMATICAL SURVEYS, 62 196-198 (2007) [C1] | | |
2007 | Zudilin VV, 'More Ramanujan-type formulae for 1/p 2', Russian Mathematical Surveys, 62 634-636 (2007) [C1] | | |
2006 | Matala-Aho T, Vaananen K, Zudilin W, 'New irrationality measures for q-logarithms', MATHEMATICS OF COMPUTATION, 75 879-889 (2006) [C1] | | |
2006 | Pilehrood KH, Pilehrood TH, Zudilin W, 'Irrationality of certain numbers that contain values of the di- and trilogarithm', MATHEMATISCHE ZEITSCHRIFT, 254 299-313 (2006) [C1] | | |
2006 | Sondow J, Zudilin W, 'Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper', RAMANUJAN JOURNAL, 12 225-244 (2006) [C1] | | |
2006 | Krattenthaler C, Rivoal T, Zudilin W, 'Series hypergeometriques basiques, q-analogues des valeurs de la fonction zeta et formes modulaires', Journal of the Institute of Mathematics of Jussieu, 5 53-79 (2006) [C1] | | |
2006 | Zudilin W, 'Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values', Journal of Mathematical Sciences, 137 4673-4683 (2006) [C1]We construct simultaneous rational approximations to q-series 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/1-qn=Â¿... [more] We construct simultaneous rational approximations to q-series 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/1-qn=Â¿8n=1xqn/1-xqn L2 (x;q) = Â¿8n=1n(xq)n/1-qn= Â¿8n=1xqn/1-xqn. 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, 'Baker-type estimates for linear forms in the values of q-series', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 48 147-160 (2005) [C1] | | |
2005 | Zudilin W, 'Well-poised generation of Apery-like recursions', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 178 513-521 (2005) [C1] | | |
2005 | Zudilin WV, 'Ramanujan-type formulae and irrationality measures of some multiples of p', Sbornik Mathematics, 196 983-998 (2005) | | |
2005 | Zudilin W, 'Computing powers of two generalizations of the logarithm', Seminaire Lotharingien de Combinatoire, 53 1-6 (2005) [C1] | | |
2004 | Zudilin VV, 'Binomial sums related to rational approximations to Â¿(4)', Mathematical Notes, 75 594-597 (2004) | | |
2004 | Zudilin VV, 'The inverse legendre transform of a certain family of sequences', Mathematical Notes, 76 276-279 (2004) [C1] | | |
2004 | Zudilin W, 'Binomial sums related to rational approximations to Zeta(4)', Russian Academy of Sciences: Mathematical Notes, 75 594-597 (2004) [C1] | | |
2004 | Bundschuh P, Zudilin W, 'On theorems of Gelfond and Selberg concerning integral-valued entire functions', JOURNAL OF APPROXIMATION THEORY, 130 164-178 (2004) [C1] | | |
2004 | Zudilin W, 'Well-poised hypergeometric transformations of Euler-type multiple integrals', JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 70 215-230 (2004) [C1] | | |
2004 | Zudilin W, 'Heine's basic transform and a permutation group for q-harmonic series', ACTA ARITHMETICA, 111 153-164 (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, 16 251-291 (2004) [C1] | | |
2003 | Zudilin W, 'The hypergeometric equation and Ramanujan functions', RAMANUJAN JOURNAL, 7 435-447 (2003) [C1] | | |
2003 | Zudilin VV, 'On the Functional Transcendence of q-Zeta Values', Mathematical Notes, 73 588-589 (2003) [C1] | | |
2003 | Rivoal T, Zudilin W, 'Diophantine properties of numbers related to Catalan's constant', MATHEMATISCHE ANNALEN, 326 705-721 (2003) [C1] | | |
2003 | Zudilin VV, 'Algebraic relations for multiple zeta values', Russian Mathematical Surveys, 58 1-29 (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 47-68 (2003) [C1] | | |
2003 | Zudilin W, 'An Apery-like difference equation for Catalan's constant', ELECTRONIC JOURNAL OF COMBINATORICS, 10 (2003) [C1] | | |
2003 | Zudilin W, 'An ApÃ©ry-like difference equation for Catalan's constant', Electronic Journal of Combinatorics, 10 (2003)Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised 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 very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order 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 continued-fraction expansion for it. Similar arguments are put forward to deduce a second-order difference equation and a new continued fraction for Â¿(4) = p4/90. | | |
2003 | Zudilin W, 'Well-poised hypergeometric service for diophantine problems of zeta values', Journal de Theorie des Nombres de Bordeaux, 15 593-626 (2003) [C1] | | |
2002 | Zudilin W, 'Remarks on irrationality of q-harmonic series', MANUSCRIPTA MATHEMATICA, 107 463-477 (2002) | | |
2002 | Zudilin VV, 'Very well-poised hypergeometric series and multiple integrals', Russian Mathematical Surveys, 57 824-826 (2002) | | |
2002 | Zudilin WV, 'On the irrationality measure for a q-analogue of Â¿(2)', Sbornik Mathematics, 193 1151-1172 (2002) | | |
2002 | Zudilin W, 'Irrationality of values of the Riemann zeta function', Izvestiya Mathematics, 66 489-542 (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, 71 604-616 (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 p-adic technique. | | |
2002 | Zudilin VV, 'A third-order ApÃ©ry-like recursion for Â¿(5)', Mathematical Notes, 72 733-737 (2002) | | |
2002 | Zudilin VV, 'Diophantine problems for q-Zeta values', Mathematical Notes, 72 858-862 (2002) | | |
2001 | Zudilin VV, 'One of the eight numbers Â¿(5), Â¿(7),Â¿ , Â¿(17), Â¿(19) is irrational', Mathematical Notes, 70 426-431 (2001) | | |
2001 | Bertrand D, Zudilin W, 'Derivatives of Siegel modular forms and exponential functions', Izvestiya Mathematics, 65 659-671 (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, 61 246-248 (1997) | | |
1997 | Zudilin VV, 'Recurrent sequences and the measure of irrationality of values of elliptic integrals', Mathematical Notes, 61 657-661 (1997) | | |
1996 | Zudilin VV, 'On the algebraic structure of functional matrices of special form', Mathematical Notes, 60 642-648 (1996) | | |