We prove some finite sum identities involving reciprocals of the binomial and central binomial coefficients, as well as harmonic, Fibonacci and Lucas numbers, some of which recover previously known ...results, while the others are new.
For real numbers
p
,
q
>
1
we consider the following family of integrals:
∫
0
1
(
x
q
-
2
+
1
)
log
x
mq
+
1
x
q
+
1
d
x
and
∫
0
1
(
x
p
t
-
2
+
1
)
log
x
t
+
1
x
pt
+
1
d
x
.
We evaluate these ...integrals for all
m
∈
N
,
q
=
2
,
3
,
4
and
p
=
2
,
3
explicitly. They recover some previously known integrals. We also compute many integrals over the infinite interval
0
,
∞
)
. Applying these results, we offer many new Euler–BBP- type sums.
Parametric binomial sums involving harmonic numbers Batır, Necdet
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas,
04/2021, Letnik:
115, Številka:
2
Journal Article
Recenzirano
We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for
p
=
0
,
1
,
2
and
|
t
|
≤
1
.
∑
k
=
1
∞
H
k
-
1
t
k
k
p
n
+
k
k
and
∑
k
=
1
∞
t
k
k
p
...n
+
k
k
.
We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications.
ζ
(
n
+
1
)
=
∑
k
=
n
∞
s
(
k
,
n
)
k
k
!
,
n
=
1
,
2
,
3
,
…
.
As examples,
ζ
(
3
)
=
1
7
∑
k
=
1
∞
H
k
-
1
4
k
k
2
2
k
k
,
and
ζ
(
3
)
=
8
7
+
1
7
∑
k
=
1
∞
H
k
-
1
4
k
k
2
(
2
k
+
1
)
2
k
k
,
which are new series representations for the Apéry constant
ζ
(
3
)
.
We provide an elementary proof of the left-hand side of the following inequality and give a new upper bound for it.
n
!
x
-
(
x
-
1
/
n
+
α
)
-
n
1
n
+
1
<
(
(
-
1
)
n
-
1
ψ
(
n
)
)
-
1
(
x
)
<
n
...!
x
-
(
x
-
1
/
n
+
β
)
-
n
1
n
+
1
,
where
α
=
(
n
-
1
)
!
-
1
/
n
and
β
=
n
!
ζ
(
n
+
1
)
-
1
/
n
, which was proved in Batir (J Math Anal Appl 328:452–465,
2007
), and we prove the following inequalities for the inverse of the digamma function
ψ
.
1
log
(
1
+
e
-
x
)
<
ψ
-
1
(
x
)
<
e
x
+
1
2
,
x
∈
R
.
The proofs are based on nice applications of the mean value theorem for differentiation and elementary properties of the polygamma functions.
We aim to investigate the four types of variant Euler harmonic sums. Also, as
corollaries, we provide particular examples of our core findings, some of
whose further instances are evaluated in terms ...of basic and well-known
functions as well as certain mathematical constants. We explore relevant
linkages between our results and those of other previously established
studies. An examination of a specific case of one result shows a
relationship to series involving zeta functions, which is also a popular
area of research.
Numerous logarithmic integrals have been extensively documented in the literature. This paper presents an algorithmic evaluation of a specific class of these integrals. Our systematic approach, ...rooted in logarithmic principles, enables us to extend our findings to other cases within this family of integrals. Furthermore, we explore special cases derived from our main results, thereby enhancing the applicability and significance of our work for a wider audience of researchers.
Some complete monotonicity results for q-polygamma functions are proved. Our results extend positivity of some functions containing q-polygamma functions to complete monotonicity property. Also, we ...give two new inequalities for q-trigamma function.
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