Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus ()
1. Introduction
Euler is widely regarded as one of the greatest and most prolific mathematicians of all time [1] as everyone can contemplate when reading the list of his publications concerning plenty of mathematical notions in [2] . From this list we know that he has introduced among others the notion of function
of a variable x, the exponential notation
, the symbol i such that
, the trigonometric definition of
and
by the formula
, etc.
In the same period, Leibniz and Newton introduced the notions of infinitesimals dx, dy, the derivative
of a function and so on [3] [4] .
Later, in 1766, Lagrange succeeded Euler at the Prussian Academy of Sciences and invented the shift operator
having the property
[5] .
After the introduction of shift or translation operators, it was introduced about in 1970 the notion of hyper-differential operator, noticeably by Wolf of Universidad Nacional Autónoma de México [6] who obtained that the Fourier transformation is representable by the operators
where
designates the Eckaert “multiply with x” operator analogous with the position operator in quantum mechanics.
At the same period, many authors such as Moshinsky, Quesne [7] and Treves [8] introduced the notion of linear canonical transforms.
Now, this work aims to utilize the Lagrange shift operator
to express
as the transform of the monomial
by a hyper differential operator then to obtain by differential calculus, its known and new found properties, its recursion relations; maybe all of its relations with the Bernoulli polynomials; its Fourier series which give
from series in
as so as expansions of functions into Euler series; etc.
2. Definition
The Euler polynomial is defined from the generating function [9] [10]
(1)
Utilizing the Lagrange translation operator
(2)
that haves the properties
(3)
(4)
(5)
(6)
we may write
as the transform of the monomial
under a differential operator
(7)
i.e., that
is an Appell type polynomial [11] .
3. Properties of
3.1. From (7) We Get Immediately
(8)
and the interesting property
(9)
saying that
·
is equal to the mean value of
and
.
3.2. Power and Alternating Powers Sums
From the property (9) one gets immediately by addition lines-lines
the interesting formulae on alternating sums of powers from
to
(10)
and on sums on powers from
to
(11)
As simple examples one has the remarkable identities
(12)
from which we get the formula
(13)
that one may find in [12] and the followings
(14)
3.3. Values of
From the differential form (7) of
and the property
(15)
as so as the formula for a sum of terms of a geometric progression with common ratio identical to
(16)
or, replacing n with 2n + 1,
one gets from (7)
(17)
i.e.,
a new property of Euler polynomials
(18)
For example
Formula (18) means that
·
is
times the alternating sum
or, equivalently,
is
times the alternating sum
For examples
3.4. Symmetry of Euler Polynomials
From (7) we may write
(19)
so that
(20)
i.e.,
· The graph of
is symmetric with respect to the straight line
for m even and to the point
for m odd.
In particular
(21)
(22)
(23)
where
designated the Euler number of order n.
3.5. Symbolic Formula of Euler Polynomials
Now from (7) we may the write down symbolic formula
(24)
where undefined notation
is to be replaced with well defined
.
Permuting z and a in (24) we get the complementary symbolic formula
(25)
which for
leads to the formula
(26)
similar to the Lucas formula for calculating Bernoulli polynomials [12] .
Another way for obtaining symbolic formula of Euler polynomials is by remarking that as
(27)
we may write
or, symbolically, the symbolic relations
(28)
(29)
The above results are resumed in Table 1.
4. Relations between Euler and Bernoulli Polynomials
4.1. The First Relation
From the known property of the Bernoulli polynomials [12]
(30)
which makes them also of Appell type we get
i.e., according to (7) and the property of the shift operator
, the known property one may find in Ref. [13]
(31)
4.2. The Second Relation
Now, thanks to the formula obtainable from (30).
Table 1. Simple properties of Euler polynomials.
(32)
we may put the formula (31) under the more useful form
(33)
We note that formulae (31) and (33) are proven by Roman [3] by another method.
As consequence of (33) we have the very important relation linking
with
(34)
from which one can obtain easily
knowing
and vice-versa.
For example, with
4.3. The Third Relation
Searching for other relations between Euler and Bernoulli polynomials we get from (7) the relations
(35)
(36)
and
(37)
For examples:
4.4. The Fourth Relation
Moreover we find from the differential representations of Euler and Bernoulli polynomials that
(38)
i.e., another relation between Euler and Bernoulli polynomials
(39)
For examples
4.5. The Fifth Relation
More curiously concerning the interrelation between Euler and Bernoulli polynomials we discover from the relation
(40)
the symbolic relation
(41)
For example
Formula (41) suggests the great theorem which may be very useful.
· The property
holds for any two Appell type polynomials.
4.6. The Sixth Relation
Now, if we write
(42)
then apply the operator (30) on both members of it and utilize (39) we get
(43)
and another relation between
and the Bernoulli numbers
(44)
For examples
The relations between Euler and Bernoulli polynomials are summarized in Table 2.
5. Obtaining Euler Polynomials
5.1. From Values at Origin
From the symbolic formula (42)
We see that
may be obtained from the values
.
Moreover, because of (34)
they are also obtainable from Bernoulli numbers.
Finally, in a recent work on Bernoulli polynomials [12] we have obtained the famous formula giving easily all values
, says
(45)
Combining (31) with (45) we may calculate very simply all
and all
.
For examples
Table 2. Relations Euler-Bernoulli polynomials.
5.2. From Recursion Relation on Euler Polynomials
From the following identity of operators that we characterize fundamental [14]
(46)
obtainable by recurrence from the property
(47)
we obtain the new symbolic recurrence relation on Euler polynomials
(48)
i.e.,
(49)
For example, because
,
5.3. From Euler Polynomials of Sums of Arguments
Another way for obtaining recursion relation between
comes from (49) and the property
(50)
which leads to
(51)
and
i.e., because
,
,
(52)
In particular
For examples
We note that (52) is equivalent with the formula about Bernoulli polynomials found by the similar method cited in Ref. [10]
It gives rise also to the second symbolic formula concerning
(53)
to be compared with the marvellous formula (45) concerning Bernoulli numbers
and the mixed formula coming from (44)
(54)
Some examples concerning
given by (53) where
Table 3. Recursion relations on Euler polynomials.
Hereafter we summarize the recursion relations between
and
of lower orders in Table 3.
Although (53) is convenient for calculating
we would like to expose hereafter a formula for calculating them not by recurrence but individually.
5.4. Obtaining
from
From the property (8) one may write
and by taking primitives of both sides
(55)
From (55) we get
But according to (20)
so that from
one gets after all the formula giving
from an integral of
(56)
This new algorithm may be utilized to calculate them as shown in Table 4.
· We remark that because
a similar formula as (56) for Bernoulli numbers and polynomials doesn’t exist.
For examples, noting that
,
Table 4.
and Euler polynomials
.
· Another algorithm for obtaining
is:
Let
and
be the primitive of
we see that
6. Integral of Product of Euler Polynomials
Thanks to the property (8) one may perform successive integrations by parts on products of Euler polynomials and get
But
and
so that we finally get
(57)
In particular for
(58)
Thanks to the relation (34) we see that the formula (58) is conformed with the result given in [13] says
(59)
We observe that by integration by parts one may also calculate
.
7. Fourier Series of Euler Polynomials
From the famous relation (33) between Euler and Bernoulli polynomials
and the Hurwitz formula on Bernoulli polynomials [12]
we get directly the Fourier series expansions of Euler polynomials
(60)
i.e.,
(61)
and
(62)
For example
The formulae (61), (62) were known for example in [10] . They do not depend On
or
. On the contrary they show the apparition of
and permit to calculate them by summations of infinite series, for examples
(63)
(64)
8. Euler Polynomials and Euler Zeta Functions
From the Hurwitz formula on Bernoulli polynomials [13]
we get
(65)
Formula (65) leads to the relation of the Euler-Riemann zeta function [15] with Bernoulli numbers
(66)
and, thanks to (34), with
(67)
For example
(68)
a result that Euler had proven in 1734 by a laborious method described in [15] .
More generally, by putting
in (65) we get the general formulae permitting to calculate the values of
from the values of
and
(69)
(70)
Explicitly, from (69) we get Table 5.
Regarding these results one may say that
· There have five types of infinite sums over
for calculating each
.
Besides, from (70) we get
from summation of series on
(71)
Table 5. Series of Infinite Sums over
.
and so all.
9. Euler Series of Functions
We already know the formulae on Fourier series (60)
and that entire functions may be expanded into series of Bernoulli polynomials
(72)
for examples
(73)
Now from the relations
we see that entire functions may also be expanded into series of Bernoulli as so as into series of Eulerpoly nomials.
For examples, because
we may write
(74)
(75)
(76)
The above relations may be resumed in Table 6.
10. Remarks and Conclusions
The main particularity of this work is the use of the Lagrange translation or shift Operator
that is curiously let apart by quasi all authors although this is seen here to be very useful and easy to utilize. From it, Euler polynomials
may be presented under the form of an Appell type polynomial which gives directly many algebraic properties concerning
,
,
, many relations with sums of powers, many known and new relations with the Bernoulli polynomials
, noticeably
; the symbolic relation
, a formula simultaneously giving
and
; relations between
and Euler-Riemann zeta
Table 6. Series of and on Euler polynomials.
functions
as so as between
and series on
,
generalizing
are given.
Last but not least, Fourier series of Euler polynomials and Euler series of functions are discussed and shown.
We think that this work has some significative value for the comprehension of the Euler polynomials. Nevertheless it may be completed with many works on them, one may find in literature for example by Vergara-Hermosilla [16] concerning the properties of Hurwitz polynomials and by Ghisa, D. [17] concerning Euler Product Dirichlet Functions
Acknowledgements
The author specially dedicates this works to Profs. Tu Ngoc Tinh, Nguyen Chung Tu who teach him in the years 60’s of the last century in Vietnam; to his Prof. Demeur M. and Drs. Quesne C., Reidemeister G., Deenen J., Beart G. at ULB, Brussels; Profs. Dagonnier R., Van Praag P. at UEM, Mons.
He would like also to thank very, very much his lovely wife Truong K.Q. for the perseverant cares she devote to him during his researcher life and life.
Notations
≡ Appel-type polynomials;
= Bernoulli polynomials;
= Bernoulli numbers;
= Euler polynomials;
= Euler numbers;
Symbolic relation
where
;
= Powers sums;
= Alternating sums of powers;
= Euler-Riemann zeta function.