We will tackle a conjecture of S. Seo and A. J. Yee, which says that the series expansion of 1/(q,−q3;q4)∞ has nonnegative coefficients. Our approach relies on an approximation of the generally ...nonmodular infinite product 1/(qa;qM)∞, where M is a positive integer and a is any of 1,2,…,M.
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On the Fréchet space of entire functions H(C), we show that every nonscalar continuous linear operator L:H(C)→H(C) which commutes with differentiation has a hypercyclic vector f(z) in the form of the ...infinite product of linear polynomials:f(z)=∏j=1∞(1−zaj), where each aj is a nonzero complex number.
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Let (a(n):n∈N0) denote an automatic sequence. Previous research on infinite products involving automatic sequences has mainly dealt with identities for products as in ∏nR(n)a(n) for rational ...functions R(n). This inspires the development of techniques for evaluating ∏nf(n)a(n) more generally, for functions f(n) that are not rational functions. This leads us to apply Euler's product expansion for the Γ-function together with recursive properties of a(n) to obtain identities as in ∏nf(n,z)a(n)=Γ(z+1), and this is motivated by how the equivalent series identity ∑na(n)lnf(n,z)=lnΓ(z+1) could be applied in relation to the remarkable results due to Gosper on the integration of lnΓ(z+1). We succeed in applying this approach, using Gosper's integration identities, to obtain new infinite products that we evaluate in terms of the Glaisher–Kinkelin constant A and that involve the Thue–Morse sequence, the period-doubling sequence, and the regular paperfolding sequence. A byproduct of our method gives us a way of generalizing a Dirichlet series identity due to Allouche and Sondow, and we also explore applications related to a product evaluation due to Gosper involving A.
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In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result ...and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory.
We present several inequalities for the Ramanujan generalized modular equation function
μ
a
(
r
)
=
π
F
(
a
,
1
-
a
;
1
;
1
-
r
2
)
/
2
sin
(
π
a
)
F
(
a
,
1
-
a
;
1
;
r
2
)
with
a
∈
(
0
,
1
/
2
...and
r
∈
(
0
,
1
)
, and provide an infinite product formula for
μ
1
/
4
(
r
)
, where
F
(
a
,
b
;
c
;
x
)
=
2
F
1
(
a
,
b
;
c
;
x
)
is the Gaussian hypergeometric function.
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In this paper, we construct several double sequences arising from double integrals which can be interpreted as alternating sums of multiple zeta values. The aim of the article is based on generating ...functions and an infinite product representation for the gamma function to find relations among alternating sums of multiple zeta values. As a consequence, we give an approach to obtain an evaluation of the multiple zeta-star values ζ⋆(m+2,{2}n).
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9.
Turán type inequalities for Struve functions Baricz, Árpád; Ponnusamy, Saminathan; Singh, Sanjeev
Journal of mathematical analysis and applications,
01/2017, Volume:
445, Issue:
1
Journal Article
Peer reviewed
Open access
Some Turán type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like ...Mittag–Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modified Struve functions of the first and second kind.
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We explore the evaluation of infinite products involving the automatic sequence
(
d
n
:
n
∈
N
0
)
known as the period-doubling sequence, inspired by the work of Allouche, Riasat, and Shallit on the ...evaluation of infinite products involving the Thue–Morse or Golay–Shapiro sequences. Our methods allow for the application of integral operators that result in new product expansions for expressions involving the dilogarithm function, resulting in new formulas involving Catalan’s constant
G
, such as the formula
∏
n
=
1
∞
n
+
2
n
n
+
1
4
n
+
3
4
n
+
5
4
n
+
4
d
n
=
e
2
G
π
2
introduced in this article. More generally, the evaluation of infinite products of the form
∏
n
=
1
∞
e
(
n
)
d
n
for an elementary function
e
(
n
) is the main purpose of our article. Past work on infinite products involving automatic sequences has mainly concerned products of the form
∏
n
=
1
∞
R
(
n
)
a
(
n
)
for an automatic sequence
a
(
n
) and a rational function
R
(
n
), in contrast to our results as in above displayed product evaluation. Our methods also allow us to obtain new evaluations involving
ζ
(
3
)
π
2
for infinite products involving the period-doubling sequence.
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