,Preface
As many of you may easily guess after reading the title of the book, the present
work is the sequel to my first book, (Almost) Impossible Integrals, Sums, and Series,
published by Springer in 2019, or to put it simply, we may view it as the second
volume of it. The title More (Almost) Impossible Integrals, Sums, and Series comes
from an old discussion with Paul Nahin, the author of the famous book Inside
Interesting Integrals, at the end of 2015, when he suggested me to also consider the
possibility of writing a second title (at that time, I didn’t know how the publisher
would react to my first book proposal, but we both talked about these matters with a
positive thinking and expecting a good outcome). Fortunately, the course of my first
book project was a good one and eventually got published.
The thing that has played a major motivational part and has given me all the
necessary stamina for the continuation of the work for another book has been Many
of You, Dear Readers!, your positive reactions I have received after the publication
day of my first book, May 11, 2019. A few days after that day, I received more
messages highlighting more or less directly that it would be nice if I continued
writing such books, which, as time passed by, made all crystal clear to me that it is
a good idea to go on and write a second book.
Writing a book is one thing, but giving the proper soul to your book, in order to
conquer the hearts of your readers with the beauty of the mathematical results and
calculations in it, is another thing, and so important and challenging, a point which
I have always tried to carefully consider.
As my first book, (Almost) Impossible Integrals, Sums, and Series, the present
book is dominated by a strong influence of the harmonic number world. If in my first
book I focused on the harmonic series with a classical structure, in this work, special
attention and treatment will be given to the atypical harmonic series, especially the
(m) (m)
ones involving harmonic numbers of the type .(H2n )p , .(H n )p , where the last one
is known, in simple terms, as a skew-harmonic number.
Let me now use the power of the examples, and I’ll start from the famous
∞
Hn 2 17
quadratic series of Au-Yeung, . = ζ (4), which I calculated in my
n 4
n=1
vii
,viii Preface
first book title. What if I now used .Hn H2n instead of .Hn2 ? And what if I also
modified the denominator and used .(2n + 1)2 instead of .n2 ? You might find that
such modifications almost bring us in a different world of calculations with different
challenges, often difficult or very difficult challenges!
Here are the versions in closed form I just talked about:
∞
Hn H2n
.
n2
n=1
13 7 1 1
. = ζ (4) + log(2)ζ (3) − log2 (2)ζ (2) + log4 (2) + 4 Li4
8 2 6 2
and
∞
Hn H2n
.
(2n + 1)2
n=1
1 1 7 1 1
. = log4 (2) − log2 (2)ζ (2) + log(2)ζ (3) − ζ (4) + 2 Li4 .
12 2 8 4 2
Very different closed forms when compared to the one of Au-Yeung series given
earlier! And with some courage, we may also take a look at the more advanced
versions of the series above I recently obtained in my research and that were
published in JCA (Journal of Classical Analysis), this time with a weight 5
structure:
∞
Hn H2n
.
n3
n=1
307 1 8 8
. = ζ (5) − ζ (2)ζ (3) − 7 log2 (2)ζ (3) + log3 (2)ζ (2) − log5 (2)
16 2 3 15
1 1
. − 16 log(2) Li4 − 16 Li5
2 2
and
∞
Hn H2n
.
(2n + 1)3
n=1
1 1 7 17 31
. = log5 (2) − log3 (2)ζ (2) + log2 (2)ζ (3) − log(2)ζ (4) + ζ (5)
12 2 4 8 128
1
. + 2 log(2) Li4 .
2
, Preface ix
And then, since we are talking about harmonic series with a weight 5 structure, it
is also worth mentioning the ones with summands involving the generalized skew-
harmonic numbers! Here are two splendid examples:
∞
Hn H n
.
n3
n=1
1 7 193 3
. = log3 (2)ζ (2) − log2 (2)ζ (3) + 4 log(2)ζ (4) − ζ (5) + ζ (2)ζ (3)
6 8 64 8
1 1
.− log5 (2) + 2 Li5
60 2
and then
∞ (2)
Hn H n
.
n2
n=1
29 5 7 2 2
. = ζ (5) − ζ (2)ζ (3) + log2 (2)ζ (3) − log3 (2)ζ (2) + log5 (2)
64 4 4 3 15
1 1
. + 4 log(2) Li4 + 4 Li5 .
2 2
Observe that so far the atypical harmonic series presented above as examples
have been all non-alternating! Let’s jump now to the atypical alternating ones!
Have you ever encountered the alternating harmonic series of weight 4,
∞
Hn
. (−1)n−1 3
n
n=1
11 7 1 1 1
. = ζ (4) − log(2)ζ (3) + log2 (2)ζ (2) − log4 (2) − 2 Li4 ,
4 4 2 12 2
which is also presented in my first book title? People interested in the world of
harmonic series will (inevitably) meet it one day, and they might find it challenging
to some extent! And if someone intends to find an elegant way to calculate it, then
the difficulty will increase! An elegant solution to this alternating harmonic series,
one that exploits two beta function representations, may be found in the present
book. Having said that, let’s imagine now we replace .Hn by .H2n in the previous
alternating harmonic series. What closed form would we obtain? Well, as we’ll see
later in the book, we’ll (surprisingly) find that
As many of you may easily guess after reading the title of the book, the present
work is the sequel to my first book, (Almost) Impossible Integrals, Sums, and Series,
published by Springer in 2019, or to put it simply, we may view it as the second
volume of it. The title More (Almost) Impossible Integrals, Sums, and Series comes
from an old discussion with Paul Nahin, the author of the famous book Inside
Interesting Integrals, at the end of 2015, when he suggested me to also consider the
possibility of writing a second title (at that time, I didn’t know how the publisher
would react to my first book proposal, but we both talked about these matters with a
positive thinking and expecting a good outcome). Fortunately, the course of my first
book project was a good one and eventually got published.
The thing that has played a major motivational part and has given me all the
necessary stamina for the continuation of the work for another book has been Many
of You, Dear Readers!, your positive reactions I have received after the publication
day of my first book, May 11, 2019. A few days after that day, I received more
messages highlighting more or less directly that it would be nice if I continued
writing such books, which, as time passed by, made all crystal clear to me that it is
a good idea to go on and write a second book.
Writing a book is one thing, but giving the proper soul to your book, in order to
conquer the hearts of your readers with the beauty of the mathematical results and
calculations in it, is another thing, and so important and challenging, a point which
I have always tried to carefully consider.
As my first book, (Almost) Impossible Integrals, Sums, and Series, the present
book is dominated by a strong influence of the harmonic number world. If in my first
book I focused on the harmonic series with a classical structure, in this work, special
attention and treatment will be given to the atypical harmonic series, especially the
(m) (m)
ones involving harmonic numbers of the type .(H2n )p , .(H n )p , where the last one
is known, in simple terms, as a skew-harmonic number.
Let me now use the power of the examples, and I’ll start from the famous
∞
Hn 2 17
quadratic series of Au-Yeung, . = ζ (4), which I calculated in my
n 4
n=1
vii
,viii Preface
first book title. What if I now used .Hn H2n instead of .Hn2 ? And what if I also
modified the denominator and used .(2n + 1)2 instead of .n2 ? You might find that
such modifications almost bring us in a different world of calculations with different
challenges, often difficult or very difficult challenges!
Here are the versions in closed form I just talked about:
∞
Hn H2n
.
n2
n=1
13 7 1 1
. = ζ (4) + log(2)ζ (3) − log2 (2)ζ (2) + log4 (2) + 4 Li4
8 2 6 2
and
∞
Hn H2n
.
(2n + 1)2
n=1
1 1 7 1 1
. = log4 (2) − log2 (2)ζ (2) + log(2)ζ (3) − ζ (4) + 2 Li4 .
12 2 8 4 2
Very different closed forms when compared to the one of Au-Yeung series given
earlier! And with some courage, we may also take a look at the more advanced
versions of the series above I recently obtained in my research and that were
published in JCA (Journal of Classical Analysis), this time with a weight 5
structure:
∞
Hn H2n
.
n3
n=1
307 1 8 8
. = ζ (5) − ζ (2)ζ (3) − 7 log2 (2)ζ (3) + log3 (2)ζ (2) − log5 (2)
16 2 3 15
1 1
. − 16 log(2) Li4 − 16 Li5
2 2
and
∞
Hn H2n
.
(2n + 1)3
n=1
1 1 7 17 31
. = log5 (2) − log3 (2)ζ (2) + log2 (2)ζ (3) − log(2)ζ (4) + ζ (5)
12 2 4 8 128
1
. + 2 log(2) Li4 .
2
, Preface ix
And then, since we are talking about harmonic series with a weight 5 structure, it
is also worth mentioning the ones with summands involving the generalized skew-
harmonic numbers! Here are two splendid examples:
∞
Hn H n
.
n3
n=1
1 7 193 3
. = log3 (2)ζ (2) − log2 (2)ζ (3) + 4 log(2)ζ (4) − ζ (5) + ζ (2)ζ (3)
6 8 64 8
1 1
.− log5 (2) + 2 Li5
60 2
and then
∞ (2)
Hn H n
.
n2
n=1
29 5 7 2 2
. = ζ (5) − ζ (2)ζ (3) + log2 (2)ζ (3) − log3 (2)ζ (2) + log5 (2)
64 4 4 3 15
1 1
. + 4 log(2) Li4 + 4 Li5 .
2 2
Observe that so far the atypical harmonic series presented above as examples
have been all non-alternating! Let’s jump now to the atypical alternating ones!
Have you ever encountered the alternating harmonic series of weight 4,
∞
Hn
. (−1)n−1 3
n
n=1
11 7 1 1 1
. = ζ (4) − log(2)ζ (3) + log2 (2)ζ (2) − log4 (2) − 2 Li4 ,
4 4 2 12 2
which is also presented in my first book title? People interested in the world of
harmonic series will (inevitably) meet it one day, and they might find it challenging
to some extent! And if someone intends to find an elegant way to calculate it, then
the difficulty will increase! An elegant solution to this alternating harmonic series,
one that exploits two beta function representations, may be found in the present
book. Having said that, let’s imagine now we replace .Hn by .H2n in the previous
alternating harmonic series. What closed form would we obtain? Well, as we’ll see
later in the book, we’ll (surprisingly) find that
