Tatiana Hessami Pilehrood - Profile on Academia.edu (original) (raw)

Papers by Tatiana Hessami Pilehrood

arXiv (Cornell University), Jan 10, 2008

Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and Bailey-Borwein-Bradl... more Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and Bailey-Borwein-Bradley's identities for generating functions of the sequences {ζ(2n+ 2)} n≥0 , {ζ(2n + 3)} n≥0 . By the same method we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function.

Journal of Mathematical Analysis and Applications, May 1, 2017

We give a new proof of Zagier's theorem on evaluation of multiple zeta (star) values at ({2} a , ... more We give a new proof of Zagier's theorem on evaluation of multiple zeta (star) values at ({2} a , 3, {2} b ). The novelty of the proof consists in employing a hypergeometric identity of Andrews, which expresses a very-well-poised hypergeometric series in terms of a multiple series, and the residue theorem.

Mathematical Notes, Mar 1, 2005

In this paper, we use Hermite-Padé approximations of the second kind to obtain a lower bound for ... more In this paper, we use Hermite-Padé approximations of the second kind to obtain a lower bound for the absolute value of a linear form, with integer coefficients, in values of polylogarithmic functions at a rational point. This estimate takes into account the growth of all coefficients of the linear form.

Journal of Number Theory, Mar 1, 2007

The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the ... more The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the principles of Chudnovsky and then apply them to obtain conditional irrationality measures for values of the digamma function.

arXiv (Cornell University), Aug 21, 2007

We study a problem of finding good approximations to Euler's constant γ = lim n→∞ S n , where S n... more We study a problem of finding good approximations to Euler's constant γ = lim n→∞ S n , where S n = n k=1 1 n -log(n + 1), by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence S n can be significantly improved if S n is replaced by linear combinations of S n with integer coefficients. In this paper, considering more general linear transformations of the sequence S n we establish new accelerating convergence formulae for γ. Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.

arXiv (Cornell University), Jan 22, 2008

By application of the Markov-WZ method, we prove a more general form of a bivariate generating fu... more By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n + 4m + 3), n, m ≥ 0, convergent at the geometric rate with ratio 2 -10 .

Discrete Mathematics & Theoretical Computer Science

We prove generating function identities producing fast convergent series for the sequences beta(2... more We prove generating function identities producing fast convergent series for the sequences beta(2n + 1); beta(2n + 2) and beta(2n + 3), where beta is Dirichlet's beta function. In particular, we obtain a new accelerated series for Catalan's constant convergent at a geometric rate with ratio 2(-10); which can be considered as an analog of Amdeberhan-Zeilberger's series for zeta(3)

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence S n = ( 6n 3n )( 3n 2n ) , and the binomial coefficients 3n n and 4n 2n . As an application, we address several conjectures of Z. W. Sun on congruences of sums involving S n and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3n n conjectured by Z. H. Sun.

Journal of the Mathematical Society of Japan, 2021

We mainly answer two open questions about finite multiple harmonic q-series on 3-2-1 indices at r... more We mainly answer two open questions about finite multiple harmonic q-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.

Journal of Mathematical Analysis and Applications, 2014

In this paper we study properties of a q-analogue of the function π / sin π z which is defined by... more In this paper we study properties of a q-analogue of the function π / sin π z which is defined by means of Jackson's q-gamma function and a reflection formula for the gamma function. As application, we obtain evaluations for q-analogues of double Euler sums in terms of single q-zeta values. We also derive a q-analogue of a well-known formula for the multiple zeta-star value ζ * ({2} a).

On Sums of Two Rational Cubes

Proceedings of Indian National Science Academy, 2015

ON SUMS OF TWO RATIONAL CUBES

arXiv: Number Theory, Oct 7, 2010

Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Eu... more Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant γ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's approximations in terms of Meijer G-functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and γ with rational coefficients. Using this construction we find new rational approximations to γ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Euler's constant. It seems to be the first non-trivial continued fraction expansion convergent to Euler's constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler-Gompertz constant found by Stieltjes in 1895. m n+m m and satisfy the second-order linear recurrence relation (n + 1) 3 y n+1 − (34n 3 + 51n 2 + 27n + 5)y n + n 3 y n−1 = 0

Generating function identities for (2n + 2); (2n + 3) via the WZ method

Electr J Comb, 2008

The Ramanujan Journal, 2017

We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of ... more We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of arbitrary depth with strings of indices composed of ones, twos and threes. Most of these results are q-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple q-harmonic sums.

Series acceleration formulae for beta values

Discrete Mathematics & Theoretical Computer Science, 2010

We generalize well-known Catalan-type integrals for Euler’s constant to values of the generalized... more We generalize well-known Catalan-type integrals for Euler’s constant to values of the generalized Euler constant function and its derivatives. Using generating functions appearing in these integral representations, we give new Vacca and Ramanujan-type series for values of the generalized Euler constant function and Addison-type series for values of the generalized Euler constant function and its derivative. As a consequence, we get base-B rational series for log 4 G π π (where G is Catalan’s constant), ζ ′ (2) π2 and also for logarithms of the Somos and Glaisher-Kinkelin constants. 1

Irrationality of certain numbers that contain values of the di- and trilogarithm

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1),S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1), and the binomial coefficients 3nchoosen{3n\choose n}3nchoosen and 4nchoose2n{4n\choose 2n}4nchoose2n. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving SnS_nSn and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3nchoosen{3n\choose n}3nchoosen conjectured by Z.\ H.\ Sun.

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1),S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1), and the binomial coefficients 3nchoosen{3n\choose n}3nchoosen and 4nchoose2n{4n\choose 2n}4nchoose2n. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving SnS_nSn and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3nchoosen{3n\choose n}3nchoosen conjectured by Z.\ H.\ Sun.

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1),S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1), and the binomial coefficients 3nchoosen{3n\choose n}3nchoosen and 4nchoose2n{4n\choose 2n}4nchoose2n. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving SnS_nSn and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3nchoosen{3n\choose n}3nchoosen conjectured by Z.\ H.\ Sun.

arXiv (Cornell University), Jan 10, 2008

Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and Bailey-Borwein-Bradl... more Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and Bailey-Borwein-Bradley's identities for generating functions of the sequences {ζ(2n+ 2)} n≥0 , {ζ(2n + 3)} n≥0 . By the same method we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function.

Journal of Mathematical Analysis and Applications, May 1, 2017

We give a new proof of Zagier's theorem on evaluation of multiple zeta (star) values at ({2} a , ... more We give a new proof of Zagier's theorem on evaluation of multiple zeta (star) values at ({2} a , 3, {2} b ). The novelty of the proof consists in employing a hypergeometric identity of Andrews, which expresses a very-well-poised hypergeometric series in terms of a multiple series, and the residue theorem.

Mathematical Notes, Mar 1, 2005

In this paper, we use Hermite-Padé approximations of the second kind to obtain a lower bound for ... more In this paper, we use Hermite-Padé approximations of the second kind to obtain a lower bound for the absolute value of a linear form, with integer coefficients, in values of polylogarithmic functions at a rational point. This estimate takes into account the growth of all coefficients of the linear form.

Journal of Number Theory, Mar 1, 2007

The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the ... more The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the principles of Chudnovsky and then apply them to obtain conditional irrationality measures for values of the digamma function.

arXiv (Cornell University), Aug 21, 2007

We study a problem of finding good approximations to Euler's constant γ = lim n→∞ S n , where S n... more We study a problem of finding good approximations to Euler's constant γ = lim n→∞ S n , where S n = n k=1 1 n -log(n + 1), by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence S n can be significantly improved if S n is replaced by linear combinations of S n with integer coefficients. In this paper, considering more general linear transformations of the sequence S n we establish new accelerating convergence formulae for γ. Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.

arXiv (Cornell University), Jan 22, 2008

By application of the Markov-WZ method, we prove a more general form of a bivariate generating fu... more By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n + 4m + 3), n, m ≥ 0, convergent at the geometric rate with ratio 2 -10 .

Discrete Mathematics & Theoretical Computer Science

We prove generating function identities producing fast convergent series for the sequences beta(2... more We prove generating function identities producing fast convergent series for the sequences beta(2n + 1); beta(2n + 2) and beta(2n + 3), where beta is Dirichlet's beta function. In particular, we obtain a new accelerated series for Catalan's constant convergent at a geometric rate with ratio 2(-10); which can be considered as an analog of Amdeberhan-Zeilberger's series for zeta(3)

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence S n = ( 6n 3n )( 3n 2n ) , and the binomial coefficients 3n n and 4n 2n . As an application, we address several conjectures of Z. W. Sun on congruences of sums involving S n and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3n n conjectured by Z. H. Sun.

Journal of the Mathematical Society of Japan, 2021

We mainly answer two open questions about finite multiple harmonic q-series on 3-2-1 indices at r... more We mainly answer two open questions about finite multiple harmonic q-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.

Journal of Mathematical Analysis and Applications, 2014

In this paper we study properties of a q-analogue of the function π / sin π z which is defined by... more In this paper we study properties of a q-analogue of the function π / sin π z which is defined by means of Jackson's q-gamma function and a reflection formula for the gamma function. As application, we obtain evaluations for q-analogues of double Euler sums in terms of single q-zeta values. We also derive a q-analogue of a well-known formula for the multiple zeta-star value ζ * ({2} a).

On Sums of Two Rational Cubes

Proceedings of Indian National Science Academy, 2015

ON SUMS OF TWO RATIONAL CUBES

arXiv: Number Theory, Oct 7, 2010

Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Eu... more Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant γ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's approximations in terms of Meijer G-functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and γ with rational coefficients. Using this construction we find new rational approximations to γ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Euler's constant. It seems to be the first non-trivial continued fraction expansion convergent to Euler's constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler-Gompertz constant found by Stieltjes in 1895. m n+m m and satisfy the second-order linear recurrence relation (n + 1) 3 y n+1 − (34n 3 + 51n 2 + 27n + 5)y n + n 3 y n−1 = 0

Generating function identities for (2n + 2); (2n + 3) via the WZ method

Electr J Comb, 2008

The Ramanujan Journal, 2017

We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of ... more We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of arbitrary depth with strings of indices composed of ones, twos and threes. Most of these results are q-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple q-harmonic sums.

Series acceleration formulae for beta values

Discrete Mathematics & Theoretical Computer Science, 2010

We generalize well-known Catalan-type integrals for Euler’s constant to values of the generalized... more We generalize well-known Catalan-type integrals for Euler’s constant to values of the generalized Euler constant function and its derivatives. Using generating functions appearing in these integral representations, we give new Vacca and Ramanujan-type series for values of the generalized Euler constant function and Addison-type series for values of the generalized Euler constant function and its derivative. As a consequence, we get base-B rational series for log 4 G π π (where G is Catalan’s constant), ζ ′ (2) π2 and also for logarithms of the Somos and Glaisher-Kinkelin constants. 1

Irrationality of certain numbers that contain values of the di- and trilogarithm

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1),S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1), and the binomial coefficients 3nchoosen{3n\choose n}3nchoosen and 4nchoose2n{4n\choose 2n}4nchoose2n. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving SnS_nSn and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3nchoosen{3n\choose n}3nchoosen conjectured by Z.\ H.\ Sun.

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1),S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1), and the binomial coefficients 3nchoosen{3n\choose n}3nchoosen and 4nchoose2n{4n\choose 2n}4nchoose2n. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving SnS_nSn and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3nchoosen{3n\choose n}3nchoosen conjectured by Z.\ H.\ Sun.

J. Integer Seq., 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be prov... more In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1),S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},Sn=frac6nchoose3n3nchoose2n22nchoosen(2n+1), and the binomial coefficients 3nchoosen{3n\choose n}3nchoosen and 4nchoose2n{4n\choose 2n}4nchoose2n. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving SnS_nSn and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3nchoosen{3n\choose n}3nchoosen conjectured by Z.\ H.\ Sun.