In a recent study of representing Dirac's delta distribution using q-exponentials, Jauregui and Tsallis experimentally discovered formulae for π as hypergeometric series as well as certain integrals. Herein, we offer rigorous proofs of these identities using various methods and our primary intent is to lay down an illustration of the many technical underpinnings of such evaluations. This includes an explicit discussion of creative telescoping and Carlson's Theorem. We also generalize the Jauregui–Tsallis identities to integrals involving Chebyshev polynomials. In our pursuit, we provide an interesting tour through various topics from classical analysis to the theory of special functions.
REFERENCES
1.
M.
Abramowitz
and I.
Stegun
, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables
(Dover
, New York
, 1965
).2.
G. E.
Andrews
, R.
Askey
, and R.
Roy
, Special Functions
(Cambridge University Press
, Cambridge
, 1999
).3.
F. D.
Carlson
, “Sur une classe de séries de Taylor
,” Ph.D. dissertation (Uppsala University
, Sweden, 1914
).4.
A.
Chevreuil
, A.
Plastino
, and C.
Vignat
, “On a conjecture about Dirac's delta representation using q-exponentials
,” J. Math. Phys.
51
(9
), 093502
(2010
).5.
I. S.
Gradshteyn
and I. M.
Ryzhik
, in Table of Integrals, Series, and Products
, 7th ed., edited by A.
Jeffrey
and D.
Zwillinger
(Academic
, New York
, 2007
).6.
M.
Jauregui
and C.
Tsallis
, “New representations of π and Dirac delta using the nonextensive-statistical-mechanics q-exponential function
,” J. Math. Phys.
51
(6
), 063304
(2010
).7.
L.
Mattner
, “Complex differentiation under the integral
,” Nieuw Arch. Wiskd., IV. Ser.
5/2
(1
), 32
–35
(2001
).8.
F. W. J.
Olver
, D. W.
Lozier
, R. F.
Boisvert
, and C. W.
Clark
, NIST Handbook of Mathematical Functions
(Cambridge University Press
, 2010
).9.
M.
Petkovsek
, H.
Wilf
, and D.
Zeilberger
, A = B.
(A. K. Peters
, Wellesley
, 1996
).10.
A.
Plastino
and M. C.
Rocca
, “A direct proof of Jauregui–Tsallis’ conjecture
,” J. Math. Phys.
52
(10
), 103503
(2010
).11.
K. R.
Stromberg
, An Introduction to Classical Real Analysis
(Wadsworth
, Florence, KY
, 1981
).12.
E.
Titchmarsh
, The Theory of Functions
, 2nd ed. (Oxford University Press
, Oxford
, 1939
).© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.
