Article
Kyungpook Mathematical Journal 2024; 64(1): 133-159
Published online March 31, 2024
Copyright © Kyungpook Mathematical Journal.
Bernoulli and Euler Polynomials in Two Variables
Claudio Pita-Ruiz
Universidad Panamericana, Facultad de Ingeniería, Augusto Rodin 498, Ciudad de México, 03920, México
e-mail : cpita@up.edu.mx
Received: August 11, 2022; Revised: October 23, 2022; Accepted: November 8, 2022
Abstract
In a previous work we studied generalized Stirling numbers of the second kind
Keywords: Bernoulli number, Euler number, Bernoulli polynomial, Euler polynomial, Generalized Stirling number
1. Introduction
Bernoulli polynomials
or by the explicit formulas [3, p. 48]
where
In a previous work [10] we studied a generalization of Stirling Numbers of the second kind (GSN, for short), denoted as
(
If
In particular, if
Two trivial examples are
We summarize some facts about GSN
-
Some values
-
The GSN
-
The GSN
where m is a non-negative integer.
-
The GSN
where n is a positive integer, and
-
The GSN
-
The GSN
2. Definitions and Preliminary Results
The relation between Bernoulli numbers and Stirling numbers of the second kind, is an old known story, that dates back to Worpitzky [13]. In fact, we have
(see [7]). In the Euler case, formula (3.3) in [6] shows Euler polynomials written in terms of Stirling numbers of the second kind. In [8] and [11] we used the GSN
(Formula (2.2) was inspired by (2.1), and (2.3) was inspired by formula (3.3) in [6] mentioned before.)
In this work we use the GSN
Explicitly, we have
Clearly we have that
and similar results for E replacing B.
Some examples are the following
From (1.6) and (1.7) we can write the following families of formulas for bivariate Bernoulli and Euler polynomials
where m is a non-negative integer (the case m=0 of (2.8) and (2.9) are (2.6) and (2.7), respectively), and
where n is a positive integer.
3. Properties
We begin this section with the bivariate version of addition formulas for Bernoulli and Euler polynomials.
Proposition 3.1. We have
where P is B or E.
Proof. In the case P=B, by using (1.5) we have
as desired. The proof of the Euler case is similar.
We have the following particular case of (3.1), corresponding to
By setting
From the case P=B of (3.2) with
In particular, with
That is, we have the identity
Formula (3.6) is the well-known Carlitz identity [1].
Similarly, we have from the case P=E of (3.2) with
By setting
Combining formulas for
Moreover, we can use Cheon's formula
where
where
Next we consider properties related to symmetries, difference equations and sums of powers for bivariate Bernoulli and Euler polynomials. We will need some preliminary results, contained in the following lemma.
Lemma 3.2. (a) For non-negative integer p and
(b) For non-negative integers
Proof. (a) We proceed by induction on p. For p=0 the result is clear. If the formula works for a given
as desired.
(b) We proceed by induction on
as desired.
Proposition 3.3. The bivariate Bernoulli and Euler polynomials have the following properties (when possible, we write P to denote B or E).
(a) (Symmetry) We have
(b) (Difference equation) We have
(c) (Sum of powers) If r is a positive integer, then
(d) (Derivatives) We have
(e) (Integrals) We have
Proof. (a) Let us consider the case P=B. In (3.2) replace
from where (3.15) follows. The proof in the case P=E is similar.
(b) By using (1.10) and (3.14) we have
as desired. Similarly, by using (1.10) we have
as claimed.
(c) The case r=1 of (3.18) is (3.16). If we suppose that (3.18) is valid for a positive integer
as desired. The proof of (3.19) is similar.
(d) Formulas (3.20) and (3.21) say that for
Similarly, we have
(e) We use (3.20) and (3.21) to write
which proves (3.22). In the last step we used (3.15) to write
An immediate consequence of (3.22) is the following:
Corollary 3.4. If
4. Some Identities
In this section we obtain some identities involving bivariate Bernoulli and/or Euler polynomials.
Proposition 4.1. We have the following identities
(1)
(2)
(3) For
(4) For
(5) For
Proof. We show the proofs of (4.1), (4.3) and (4.5).
(1) Beginning with the right-hand side of (4.1), we have
By using the identity
as desired
(2) Beginning with the left-hand side of (4.3), we have
By using the identity
as desired.
(3) Beginning with the right-hand side of (4.5), we have
By using the identity
as desired.
The proofs of (4.2) and (4.4) are similar, using the identities
(see [9, Eq. 24.4.23])
and
(see [9, Eq. 24.4.20]).
Now we consider a different kind of identities involving Bernoulli and Euler numbers and polynomials, which are based in the following result.
Proposition 4.2. The polynomial identity
implies the Bernoulli and Euler polynomials identities
Proof. Observe that the polynomial identity (4.9) comes together with the identities of the corresponding derivatives
for
which shows (4.10). The proof of (4.11) is similar.
For example, from (3.2) we can write
where y,z are arbitrary parameters. By using Proposition 4.2 in (4.13) we obtain identities of the form
where P,Q are Bernoulli or Euler polynomials.
In the case P=Q=B, with
from where we conclude that, if
With similar arguments it is possible to show that if
Now let us consider the difference equations (3.16) and (3.17). Write (3.16) as
By using Proposition 4.2, we obtain from (4.18) the identities
and
Similarly, by writing (3.17) as
and using Proposition 4.2, we obtain from (4.21) identities of the form
where P and Q are Bernoulli or Euler polynomials.
To end this section, we show some particular cases of the previous results.
By setting
Similarly, from (4.20) with
By using Proposition 4.2 in the case m=2 of (4.3), we obtain that
Set
The case
which can be written (by using the Euler's identity
With similar procedures it is possible to obtain also the identities
5. Generalized Recurrences
In this section we show some recurrences for bivariate Bernoulli and Euler polynomials.
Proposition 5.1. For non-negative integers
and
Proof. Formulas in (5.1) are included as particular cases of Proposition 4 in [12]. We prove the identity
contained in (5.2) by induction on q. The case q=0 is the definition (2.5). If we suppose that (5.3) is true for a given
The induction hypothesis gives us from (5.4), that
According to the recurrence (1.11), we have from (5.5) that
as desired. In the last step we used (1.10).
The proof of the identity
is similar.
Proposition 5.2. For non-negative integers
Proof. We prove the identity
contained in (5.7) by induction on q. The case q=0 of (5.8) is the definition (2.5). If we suppose (5.8) is true for a given
We use induction hypothesis to write (5.9) as
The recurrence (1.11) gives us
as desired. The proof of the identity
and (5.6) are similar.
We can write recurrence (5.7) as
Thus, from (5.2) and (5.10) we have
From (5.1) with
We use Carlitz identity (3.5) to obtain the following explicit enriched version of (5.12),
Similarly, from (5.6) we obtain
and from (5.11) we get
Proposition 5.3. We have the following identities for bivariate Euler polynomials
Proof. Let us prove (5.14) by induction on q. The case q=0 is a trivial identity. If (5.14) is valid for a given
The induction hypothesis gives us
as desired. In the last step we used the identity
which is included in the case q=1 of (5.2) (or (5.7)).
Let us prove (5.15) by induction on q. The case q=0 is a trivial identity. If we suppose (5.15) is valid for a given
as desired. We used (5.16) in the last step.
There are similar results to (5.14) and (5.15) in the case of bivariate Bernoulli polynomials. These results appear in [12] in the context of bivariate poly-Bernoulli polynomials.
Acknowledgements
I thank the anonymous referee for his/her careful reading of the original version of this work. The many details and observations he/she pointed out, certainly contributed to present this final version.
References
- L. Carlitz. Problem 795, Math. Mag., 44(1971), 107.
- G.-S. Cheon. A Note on the Bernoulli and Euler polynomials, Appl. Math. Lett., 16(3)(2003), 365-368.
- L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974, xi+343 pp.
- K. Dilcher and C. Vignat. General convolution identities for Bernoulli and Euler polynomials, J. Math. Anal. Appl., 435(2)(2016), 1478-1498.
- H. W. Gould. Explicit formulas for Bernoulli numbers, Amer. Math. Monthly, 79(1972), 44-51.
- B. N. Guo and F. Qi. Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272(2014), 251-257.
- B. C. Kellner. Identities between polynomials related to Stirling and harmonic numbers, Integers, 14(2014), Paper No. A54, 22 pp..
- T. Komatsu, B. K. Patel and C. Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials, Adv. Math. Commun., 17(3)(2023), 522-535.
- F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl and M. A. McClain, eds, NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.3 of 2021-09-15.
- C. Pita-Ruiz. Generalized Stirling Numbers I, arXiv:1803.05953v1 [math.CO].
- C. Pita-Ruiz. On a family of Euler type numbers and polynomials, Rend. Mat. Appl. (7), 44(2023), 1-26.
- C. Pita-Ruiz. On bi-variate poly-Bernoulli polynomials, Commun. Math., 31(1)(2023), 179-203.
- J. Worpitzky. Studien über die Bernoullischen und Eulerschen Zahlen, J. Reine Angew. Math., 94(1883), 203-232.