Selection of Coherent and Concise Formulae on Bernoulli Polynomials-Numbers-Series and Power Sums-Faulhaber Problems ()
1. Introduction
In many branches of mathematics the problem of Bernoulli numbers related to the millenary problem of power sums is probably the most studied since the publication of the book Ars Conjectandi by Euler in 1738 [1] as we can see on the net and, specially, in a didactical thesis of Coen [2] , the explicative work of Raugh [3] , Beardon [4] , the bibliography of thousands of articles on Bernoulli numbers realized by Dilcher, Shula, Slavutskii [5] , etc.
Concerning Bernoulli polynomials
, classically defined from a generating function, there had not so much properties, the most remarkable is its representation by a hyper-differential operator, the Hurwitz expansion of them into Fourier series, the Roman formula for
, the Euler-McLaurin formula, etc.
As for the power sums on real and complex numbers, including the famous Faulhaber conjecture, there has no valuable formula linking them with Bernoulli polynomials until only some years ago [6] .
Regarding the situation, we would like to perform a selection of as many as possible known and new interesting properties of Bernoulli polynomials then of Bernoulli numbers in a coherent way, i.e., by only one approach, which utilizes principally operator calculus lying on the couple of operators position and derivation, similar as the couple
in quantum mechanics.
In Section 2, we will treat the problem of Bernoulli polynomials, from their representation by a hyper-differential operator to almost all of their algebraic properties to the fact that
is equal to the primitive of the power sums of natural integers. Afterward we show that the formula giving Bernoulli polynomials of a sum of two arguments
leads to two new recurrence relations for obtaining
. We also give another approach for calculating without integrations, the Fourier series of Bernoulli polynomials and the Bernoulli series of functions, the relation of
with the Euler zeta function. Afterward we show an up-to-date procedure for obtaining
from and only from
leading to the rapid establishment of Table of Bernoulli polynomials and numbers. Finally, we show a new way for obtaining Fourier series of Bernoulli polynomials, Euler zeta function, and vice-versa, the series of functions in term of a set of Bernoulli polynomials.
In Section 3, we treat the problem of Bernoulli numbers
, from its initial definition by Jakob Bernoulli in 1713 who related them by conjecture with the power sums on natural numbers. By comparison of this relation with the preceding formula linking
with power sums, we may identify
with
then calculate
by a simple matrix method side-by-side with the method, more powerful, link with l, coming from the special recurrence formula coming from
.
In Section 4, we prove by utilizing the translation operator
, coming from the Newtonian binomial, that the power sums on complex numbers are simply related to those on natural numbers. On the other hand, we prove that they are also related very simply to Bernoulli polynomials, from that we get again the recurrence relation between Bernoulli polynomials.
Section 5 is devoted to the Faulhaber problem regarding power sums on complex numbers. Here we show that power sums on complex numbers may be calculated from sums of entire numbers somehow by writing
in function of the new argument
.
2. Bernoulli Polynomials
2.1. Definition and Principal Properties
In 1738, Euler introduced the Bernoulli polynomials
via the generating function [1]
(2.1)
which directly gives by identification
,
,
(2.2)
Utilizing the translation operator
coming from the Newtonian binomial
(2.3)
and having the property
(2.4)
(2.5)
we directly find from (2.1) that
is the transform of
via a differential operator
(2.6)
From (2.6) we get the famous known formulae
(2.7)
(2.8)
(2.9)
and the following formula which gives
as series of Bernoulli polynomials.
(2.10)
From (2.8) we get the formula given by Roman [7]
(2.11)
From (2.10) we get the formulae on relations of Bernoulli polynomials versus trigonometric functions, especially the Castellanos formula [8]
(2.12)
The formulae (2.7) and (2.8) give the important formulae
(2.13)
(2.14)
and the Taylor expansion
which may be put under symbolic form
(2.15)
where undefined symbols
are to be replaced with
.
Exploring now the inter-relations between Bernoulli polynomials.
From (2.4) and (2.7) we get the complementary of (2.15)
(2.16)
From (2.13)
(2.17)
i.e.,
“The sum of powers of order m of n first entire numbers from 0 to
, denoted by
, is equal to the simple primitive (without constant of integration) of the Bernoulli polynomial
” and vice-versa,
“The Bernoulli polynomial
is equal to the derivative of the power sums
”
As for
we see that
(2.18)
which leads to
(2.19)
i.e., to the theorem
“The graph of a Bernoulli polynomial is symmetric with respect to the axis
if m is pair and anti-symmetric if m is impair”.
Joint (2.19) with (2.9) we get the famous property [1]
(2.20)
Now, by replacing in (2.6) z with
so that
is with
we get
and the formula
(2.21)
saying that
is
times the sum of
For examples:
By replacing in (2.6) z with nz and
with
we find again the formula given by Raabe [9] in 1851
(2.22)
saying that
“
is
times the sum of
.”
For examples
2.2. Bernoulli Polynomials of Sum of Two Arguments
From the following property of operators that we characterize fundamental [10]
(2.23)
we get
Now, because
(2.24)
(2.25)
The above recurrence formula is to be compare with that given by Weisstein [11] without proof where there seems has a little mistake
From (2.25) and knowing that
we obtain another type of recurrence formula for Bernoulli polynomials
(2.26)
(2.27)
For examples, with
,
,
2.3. The Fourier Series of Bernoulli Polynomials. Euler Zeta Function. Powers of pi
By successive integrations by parts and utilizing the formula (2.13) for
we get, knowing (2.9),
(2.28)
Because of the factor
we may conclude that
for
and
has opposite sign with respect to
.
The same method also gives
(2.29)
which provides us the following formula on Fourier series of
proven by Hurwitz in 1890 by another method [10]
(2.30)
2.4. Bernoulli Series of Functions
Let
be a periodic function defined on an interval
and has the period
. For expanding
into a Fourier series of exponentials
,
(2.31)
we firstly write
and see that the second member is equal uniquely to
so that
(2.32)
The Fourier series of a function, if it exists, is then
(2.33)
To avoid integrations in the calculation, we may utilize the method of integrations by parts and get
so that we may write down the Fourier series formula
(2.34)
In the case
, jointed the preceding formula written under the form
with the Hurwitz formula we get the new and precious formula on expansion of derivable functions into series of Bernoulli polynomials
(2.35)
or
(2.36)
For examples, under matrix form
(2.37)
to be compared with
(2.38)
Formula (2.36) leads also to
(2.39)
(2.40)
As first interesting applications
(2.41)
By (2.36) we also obtain a precious recurrence formula of Bernoulli polynomials
(2.42)
i.e., under matrix form
(2.43)
which may be resolved for
and
my matrix calculus.
2.5. Obtaining
from
and Table of Bernoulli Polynomials
Integrating two times as followed the Hurwitz formula on Fourier series of Bernoulli polynomials we get
(2.44)
i.e.,
is equal to
times the primitive of
minus the double primitive of
calculated for
. The second term is so equal to
. (2.45)
This new algorithm for obtaining
from
and
is very easy to perform and may be utilized to establish Table of Bernoulli polynomials.
For examples:
This method for establishing a table of Bernoulli polynomials is extremely easier if we utilize the list of fifty Bernoulli numbers
conscientiously established by Coen [2] . For examples
(2.46)
2.6. Bernoulli Polynomials and Euler Zeta Function
From the Hurwitz formula
we get the Euler zeta function one may find references in Coen [2] and Raugh [3]
(2.47)
as so as
(2.48)
(2.49)
Moreover, by taking
in these formulae we get the known property
(2.50)
and the powers of pi.
For examples
(2.51)
(2.52)
(2.53)
(2.54)
and
etc.
3. Bernoulli Numbers
3.1. Definition and Properties
In 1713, according to Jacob Bernoulli (1655-1705), was published the list of ten first sums of powers of entire numbers [3]
(3.1)
in terms of the numbers
which are conjectured to be the same for all m
(3.2)
Afterward, the
were baptized Bernoulli numbers.
By comparison of the relation coming from (3.2)
(3.3)
with the formula coming from (2.16), (2.17)
(3.4)
we get, combining with (2.20),
(3.5)
i.e.
“The Bernoulli numbers
are equal to the values at origin of the Bernoulli polynomial
”.
3.2. Obtaining Bernoulli Numbers
The above formula (3.5) and the recurrence formula for Bernoulli polynomials (2.43) corresponding to
(3.6)
lead to that for Bernoulli numbers
(3.7)
which, knowing
, gives
according to following Table 1.
This matrix equation may be resolved by doing linear combinations over lines from the second one in order to replace them with lines containing only some non-zero numbers.
Table 1. Matrix equation for calculating Bm.
For instance, for calculating successively
we may utilize the matrix equation (Table 2).
We remark that the last line of this matrix has replaced
.
The results are
,
,
,
,
.
,
,
,
(3.8)
Another method, maybe more interesting, for establishing table of Bernoulli numbers is obtained from the formula (2.27). It is
or, symbolically,
(3.9)
For examples
Table 2. Simplified matrix equation for calculating Bm.
,
, etc. (3.10)
We see that
is a sum over only four terms
;
is over five,
over ten,
over twelve terms.
3.3. Obtaining Bernoulli Polynomials and Power Sums from Bernoulli Numbers
From the formula (2.15)
we get the symbolic Lucas formula
(3.11)
for calculating Bernoulli polynomials
from the set of Bernoulli numbers.
For examples
As for the power sums
we begin by calculating the formula coming from (2.17)
(3.12)
then take the primitives of both members.
For examples
3.4. Bernoulli Numbers and the Euler-McLaurin Formula
From the formula for expansion of derivable functions into series of Bernoulli polynomials
which leads, for periodic functions
identical to
in the interval
, to
(3.13)
we get the formula
(3.14)
analogue to the Euler-McLaurin formula one may find in [11]
For example, with
,
,
,
it is verified that
4. Obtaining Powers Sums of Real and Complex Numbers
4.1. From Power Sums of Integers
From the definition of the power sums on real and complex numbers
(4.1)
we get, by utilizing the translation operator
mentioned in (2.4),
(4.2)
and the formula for sums of geometric progressions, the compact formula
(4.3)
From (4.3) and the fact that
(4.4)
we get the symbolic formula
leading to the very interesting new formula given powers sums of complex numbers from powers sums of integers
(4.5)
where the undefined symbol
is to be replaced with the power sums on integers (2.17)
,
(4.6)
Another way, more shortly, to obtain (4.5) is by remarking that
so that
For examples
4.2. From Bernoulli Polynomials
Now, because n may go until infinity,
is well defined so that
(4.7)
On the other hand, from (2.18)
(4.8)
so that we obtain the following beautiful important formula
(4.9)
as so as the historic Jacobi conjectured formula
(4.10)
Formula (4.9) leads to the formula giving
directly from
(4.11)
i.e., to the algorithm saying that
is equal to
plus
and so all until
For examples
In particular, we get the recurrence relation between Bernoulli polynomials given by Roman [8]
(4.12)
and the well-known ancient formula of Bernoulli (1713)
(4.13)
Lastly, because of (4.10)
we get
and, by expanding functions into Bernoulli series, the formula found in Wikipedia
(4.14)
We resuming the herein-before results of calculations in following Tables (Tables 3-5).
Table 3. Obtaining
and
from
.
Table 4. Obtaining
from
.
Table 5. Obtaining
from
.
5. The Faulhaber Formulae on Power Sums of Complex Numbers
5.1. Powers Sums of Odd Order
Although the problems of powers sums and Faulhaber conjecture were treated by many authors for examples by Radermacher [12] , by Tsao in (2008) [13] , by Chen, Fu, Zhang in (2009) [14] , etc., nevertheless we would like to present hereafter one new approach about the problems.
In
let us replace the arguments z and n by
and
(5.1)
Because
(5.2)
and consequently
(5.3)
we have, regarding (4.9),
(5.4)
and the form of the formula for general power sums
(5.5)
that may be calculated by the following considerations.
From the property
(5.6)
we get, for utilization in (5.5),
(5.7)
and, finally,
(5.8)
All the problem is reduced to the calculations of
in function of Z which are not so difficult.
For examples:
,
and so all.
As corollary of the calculations of
we may state that
“All
and all
are polynomials of order k in Z”.
5.2. Faulhaber Formula for Even Power Sums
By differentiating both members of (5.7) and remarking that
,
we obtain the formula giving
(5.9)
For examples
The arrangement into polynomials with respect to (
) is immediate.
Remarks and Conclusions
We subjectively think that this work is a real and effective contribution to the knowledge of Bernoulli polynomials, Bernoulli numbers and Sums of powers of entire and complex numbers, as indicated in Introduction.
The main particularity of this work is the use of the translation or shift operator
that is curiously let apart by quasi all authors although this is seen to be very useful and easy to utilize.
By the utilization of many new properties on
such as
we easily get the new key formulae
together with
for obtaining
.
We find also the miraculous symbolic formula for calculating rapidly the Bernoulli numbers
which together with the Lucas symbolic formula
give easily
.
Afterward by a change of arguments from z into
and n into
we get the relation
which together with the proof that
and
are polynomials in Z gives simply rise to the Faulhaber form of
.
Operator calculus, which is very different from Heaviside operational calculus thus merits to be known. Moreover, it has a solid foundation and many interesting applications in the domains of Special functions, Differential equations, Fourier and other transforms, quantum mechanics [10] .
Acknowledgements
Because there are thousands of works about Bernoulli numbers and polynomials during centuries we surely have omitted to cite many references, we apologize for this and would like to receive comments from researchers in order to correct this work. Before, we thank you very much for that.
He thanks the assistant managers of Applied Mathematics, SCIRP very much noticeably Ms. Icy Yin and Ms. Nancy Ho for helping him in the procedure of publication of his former and recent works.
Finally, he reiterates his gratitude towards his lovely wife for all the cares she devoted for him during this work and also during his life.