No Access

THE ITERATED INTEGRALS OF ln(1 + xⁿ)

TEWODROS AMDEBERHAN

Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

Search for more papers by this author

CHRISTOPH KOUTSCHAN

Research Institute for Symbolic Computation, Johannes Kepler University, 4040 Linz, Austria

Search for more papers by this author

VICTOR H. MOLL

Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

Search for more papers by this author

, and

ERIC S. ROWLAND

LaCIM, UQAM, 201 Avenue du Président Kennedy, Montréal, QC H2X 3Y7, Canada

Search for more papers by this author

https://doi.org/10.1142/S1793042112500042Cited by:1 (Source: Crossref)

Abstract

For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x²), arithmetic properties of certain coefficients arising are described. Similar observations are made for ln(1 + x³).

Keywords:

AMSC: 26A09, 11A25

References

A. Apelblat , Tables of Integrals and Series ( Verlag Harry Deutsch , Thun, Frankfurt am Main , 1996 ) . Google Scholar
G. Boros, O. Espinosa and V. Moll, Integrals Transforms Spec. Funct. 14, 187 (2003), DOI: 10.1080/1065246031000072265. Web of Science, Google Scholar
M. Bronstein , Symbolic Integration I. Transcendental Functions , Algorithms and Computation in Mathematics 1 ( Springer-Verlag , 1997 ) . Google Scholar
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edn., eds. A. Jeffrey and D. Zwillinger (Academic Press, New York, 2007). Google Scholar
R. Graham , D. Knuth and O. Patashnik , Concrete Mathematics , 2nd edn. ( Addison-Wesley , Boston , 1994 ) . Google Scholar
C. Koutschan, Advanced applications of the holonomic systems approach, Ph.D. thesis, RISC, Johannes Kepler University, Linz, Austria (2009) . Google Scholar
C. Koutschan and V. Moll, Scientia 20, 93 (2011). Google Scholar
J. Liouville, J. Ec. Polyt. 14, 37 (1834). Web of Science, Google Scholar
J. Lützen , Joseph Liouville 1809–1882. Master of Pure and Applied Mathematics , Studies in the History of Mathematics and Physical Sciences 15 ( Springer-Verlag , New York , 1990 ) . Google Scholar
D. Manna and V. Moll, Math. Comp. 76, 2023 (2007), DOI: 10.1090/S0025-5718-07-01954-0. Web of Science, Google Scholar
D. Manna and V. Moll, Amer. Math. Monthly 114, 232 (2007). Web of Science, Google Scholar
D. Manna and V. Moll, Probab. Geom. Integrable Syst. 55, 201 (2008). Google Scholar
L. Medina, V. Moll and E. Rowland, Int. J. Number Theory 7(3), 623 (2011). Link, Web of Science, Google Scholar
V. Moll, Notices Amer. Math. Soc. 49, 311 (2002). Web of Science, Google Scholar
V. Moll, Notices Amer. Math. Soc. 57, 476 (2010). Web of Science, Google Scholar
M. Petkovšek, J. Symbolic Comput. 14(3), 243 (1992), DOI: 10.1016/0747-7171(92)90038-6. Web of Science, Google Scholar
M. Petkovšek, H. Wilf and D. Zeilberger, A=B (A. K. Peters, 1996) . Google Scholar
A. P. Prudnikov , Yu. A. Brychkov and O. I. Marichev , Integrals and Series ( Gordon and Breach Science Publishers , 1992 ) . Google Scholar
R. H. Risch, Trans. Amer. Math. Soc. 139, 167 (1969), DOI: 10.1090/S0002-9947-1969-0237477-8. Web of Science, Google Scholar
R. H. Risch, Bull. Amer. Math. Soc. 76, 605 (1970), DOI: 10.1090/S0002-9904-1970-12454-5. Web of Science, Google Scholar
J. F. Ritt , Integration in Finite Terms. Liouville's Theory of Elementary Functions ( Columbia University Press , New York , 1948 ) . Google Scholar
C. Schneider, Sém. Lothar. Combin. 56(B56b), 1 (2007). Google Scholar
X. Sun and V. Moll, Denominators of harmonic numbers, in preparation (2011) . Google Scholar