IPM Positions

Non Resident Researcher (non-resident), School of Mathematics (2010 - 2012 )

Past IPM Positions

Associate Researcher (non-resident), School of Mathematics (2006 - 2009) Associate Researcher (non-resident), School of Mathematics (2004 - 2006) Associate Researcher (non-resident), School of Mathematics (2002 - 2003)

Non IPM Affiliations

Associate Professor of Shahrekord University

Research Interests

Transcendental Number Theory, Diophantine Approximations, Linear Independence and Irrationality of Numbers, Hypergeometric Functions.

Present Research Project at IPM

Research program

More Information

1 Problem 1. The generalized polylogarithmic function is defined by the series Ls(a, z)= ? ? n=0 zn (n+a)s , where s ? N, a ? Q, 0 < a ? 1, |z| ? 1 for s ? 2 and |z| < 1 for s=1. Note that polylogarithmic function is connected with the many remarkable constants such as z(2k+1)=L2k+1(1,1) (Riemann zeta function values at odd points, k ? 2), G=1/4?L2(1/2, -1) (Catalan?s constant), ? ? n=1 1 n2 2n =L2(1, 1/2)=p2/12-1/2(log2)2 which are not yet known to be irrational. Almost all known results on the values of polylogarithms Ls(a, z) are proved for rational points close to zero. It is an interesting but very difficult problem to study the values at points that are far from zero, for example, on the boundary of the disc of convergence. In this connection we note Apery''s theorem (1978) that L3(1, 1)=z(3) is irrational, recent Rivoal''s theorem (2000) that there exist infinitely many irrational numbers among z(3), z(5), z(7), ?; Zudilin''s result (2001) that at least one of the four numbers z(5),z(7), z(9), z(11) is irrational, and also Hata''s theorem (1993) on the values of dilogarithm that L2(1, 1/q) is irrational for any integer q ? (-?, -5]?[7, ?), and author''s result (1999) that at least one of the numbers L3(1, 1/q), L2(1, 1/q) is irrational for any integer q ? (-?, -1]?[3, ?). Problem 2. Obtaining lower bounds for linear forms in values of generalized hypergeometric functions that depend on all of the coefficients. Particular cases of this problem were considered early by author. Problem 3. Diophantine equations.

Related Papers

1.	Kh. Hessami Pilehrood and Kh. Hessami Pilehrood Bivariate identities for values of the Hurwitz zeta function and supercongruences Electron. J. Combin. 18 (2012), #P35  [abstract]

2.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On a continued fraction expansion for Euler's constant J. Number Theory  (Accepted) [abstract]

3.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Rational approximations for values of Bell polynomials at points involving Euler s constant and zeta values Austral. Math. Soc. (Accepted) [abstract]

4.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Rational approximations for values of the digamma function and a denominators conjecture Math. Notes (Accepted) [abstract]

5.	Kh. Hessami Pilehrood and T. Hessami Pilehrood A q-analogue of the bailey-Borwein-Bradley identity Journal of Symbolic Computation 46 (2011), 699-711  [abstract]

6.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Vacca-type series for values of the generalized Euler constant function and its derivative Journal of Integer Sequences 13 (2010), 10.7.3  [abstract]

7.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Rational approximations for the quotient of gamma values Indag. Math. (N.S.) 20 (2009), 583-601  [abstract]

8.	Kh. Hessami Pilehrood and T. Hessami Pilehrood An Apery-like continued fraction for πcothπ J. Difference Equ. Appl. (Accepted) [abstract]

9.	Kh. Hessami Pilehrood and T. Hessamo Pilehrood Generating function identities for ζ(2n+2), zeta(2n+3) via the WZ method Electron. J. Combin. 15 (2008), 1-9  [abstract]

10.	T. Hessami Pilehrood and Kh. Hessami Pilehrood On a conjecture of Erdos Math. Notes 83 (2008), 281-284  [abstract]

11.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Infinite sums as linear combinations of polygamma functions Acta Arith. 130 (2007), 231-254  [abstract]

12.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Apéry-like recursion and continued fraction for π coth π ( In: Diophantische approximationen) [abstract]

13.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On conditional irrationality measures for values of the digamma fuction J. Number Theory 123 (2007), 241-253  [abstract]

14.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Arithmetical properties of some series with logarithmic coefficient Math. Z. 255 (2007), 117-131  [abstract]

15.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Irrationality of the sums of zeta values Math. Notes 79 (2006), 607-618  [abstract]

16.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On sums of two rational cubes Indian J. Pure Appl. Math. 36 (2006), 707-717  [abstract]

17.	Kh. Hessami Pilehrood and T. Hessami Pilehrood (Joint with W. Zudilin) Irrationality of certain numbers that contain values of the di- and trilogarithm Math. Z. 254 (2006), 299-313  [abstract]

18.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On the diophantine equation x2+3=pyn Indian J. Pure Appl. Math. 36 (2005), 431-439  [abstract]

19.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Lower Bounds for linear forms in values of polylogarithms Math. Notes 77 (2005), 573-579  [abstract]

20.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Criteria for irrationality of generalized Euler's constant J. Number Theory 108 (2004), 169-185  [abstract]