GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS

QIU-MING LUO AND FENG QI

Abstract. In this paper, the concepts of Euler numbers and Euler polynomials are generalized, and some basic properties are investigated.

1. Introduction

It is well-known that the Euler numbers and polynomials can be defined by the following definitions.

Definition 1.1 ([1]). The Euler numbers E_{k} are defined by the following expansion

\frac{2e^{t}}{e^{2t} + 1} = ∑_{k=0}^{∞} \frac{E_{k}}{k!} t^{k}, |t| ≤ π.  (1.1)

In [4, p. 5], the Euler numbers is defined by

\frac{2e^{t/2}}{e^{t} + 1} = sech \frac{t}{2} = ∑_{n=0}^{∞} \frac{(−1)^{n}E_{n}}{(2n)!} (\frac{t}{2})^{2n} , |t| ≤ π.  (1.2)

Definition 1.2 ([1, 4]). The Euler polynomials E_{k}(x) for x ∈ ℝ are defined by

\frac{2e^{xt}}{e^{t} + 1} = ∑_{k=0}^{∞} \frac{E_{k}(x)}{k!}t^{k}, |t| ≤ π.  (1.3)

It can also be shown that the polynomials E_{i}(t), i ∈ ℕ, are uniquely determined by the following two properties

E′_{i}(t) = iE_{i−1}(t), E_{0}(t) = 1;  (1.4)
E_{i}(t + 1) + E_{i}(t) = 2t^{i}.  (1.5)

2000 Mathematics Subject Classification. 11B68.

Key words and phrases. Euler numbers, Euler polynomials, generalization.

The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China.

This paper was typeset using AMS-L^{A}TEX.

2 Q.-M. LUO AND F. QI

Euler polynomials are related to the Bernoulli numbers. For information about Bernoulli numbers and polynomials, please refer to [1, 2, 3, 4].

In this paper, we will give some generalizations of the concepts of Euler numbers and Euler polynomials, and research their basic properties.

2. Generalizations of Euler numbers and polynomials

In this section, we give two definitions, the generalized Euler number and the generalized Euler polynomial, which generalize the concepts of Euler number and Euler polynomial. Definition 2.1. For positive numbers a, b, c, the generalized Euler numbers E_{k}(a, b, c) are defined by

\frac{2c^{t}}{b^{2t} + a^{2t}} = ∑_{k=0}^{∞} \frac{E_{k}(a, b, c)}{k!}t^{k}.  (2.1)

Definition 2.2. For any given positive numbers a, b, c and x ∈ ℝ, the generalized Euler polynomials E_{k}(x; a, b, c) are defined by

\frac{2c^{xt}}{b^{t} + a^{t}} = ∑_{k=0}^{∞} \frac{E_{k}(x; a, b, c)}{k!}t^{k}.  (2.2)

If taking a = 1, b = c = e, then Definition 1.1 and Definition 1.2 can be deduced from Definition 2.1 and Definition 2.2 respectively. Thus, the Definition 2.1 and Definition 2.2 generalize the concepts of Euler numbers and polynomials.

3. Some properties of the generalized Euler numbers

In this section, we study some basic properties of the generalized Euler numbers defined in Definition 2.1. Theorem 3.1. For positive numbers a, b, c and real number x ∈ ℝ, we have

E_{0}(a, b, c) = 1, E_{k}(1, e, e) = E_{k}, E_{k}(1, e^{1/2}, e^{x}) = E_{k}(x),  (3.1)
E_{k}(a, b, c) = 2^{k}(ln b − ln a)^{k}E_{k} (\frac{ln c − 2 ln a}{2(ln b − ln a)}) ,  (3.2)
E_{k}(a, b, c) = ∑_{j=0}^{k} \binom{k}{j}(ln b − ln a)^{j}(ln c − ln a − ln b)^{k−j}E_{j}.  (3.3)

Proof. The formulae in (3.1) follow from Definition 1.1, Definition 1.2, and Definition 2.1 easily.

By Definition 1.2 and Definition 2.1 and direct computation, we have

\frac{2c^{t}}{b^{2t} − a^{2t}} = \frac{2 exp(\frac{ln c−2 ln a}{2(ln b−ln a)} · 2t(ln b − ln a))}{exp(2t(ln b − ln a)) + 1}

= ∑_{k=0}^{∞} 2^{k}(ln b − ln a)^{k}E_{k} (\frac{ln c − 2 ln a}{2(ln b − ln a)}) \frac{t^{k}}{k!} .

(3.4)

Then, the formula (3.2) follows.

Substituting E_{k}(x) = ∑_{j=0}^{k} 2^{−j}\binom{k}{j}(x− \frac{1}{2})^{k−j}E_{j} into the formula (3.2) yields the formula (3.3). □

Theorem 3.2. For k ∈ ℕ, we have

E_{k}(a, b, c) = −\frac{1}{2} ∑_{j=0}^{k−1} \binom{k}{j} [(2 ln b − ln c)^{k−j} + (2 ln a − ln c)^{k−j}] E_{j}(a, b, c), (3.5)
E_{k}(a, b, c) = E_{k}(b, a, c),  (3.6)
E_{k}(a^{α}, b^{α}, c^{α}) = α^{k}E_{k}(a, b, c).  (3.7)

Proof. By Definition 2.1, direct calculation yields

1 = \frac{1}{2} [(\frac{b^{2}}{c} )^{t} + (\frac{a^{2}}{c} )^{t}] ∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(a, b, c)

= \frac{1}{2} ∑_{k=0}^{∞} \frac{t^{k}}{k!} [(ln \frac{b^{2}}{c} )^{k} + (ln \frac{a^{2}}{c} )^{k}] ∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(a, b, c)

= \frac{1}{2} ∑_{k=0}^{∞} (∑_{j=0}^{k} \binom{k}{j} [(ln \frac{b^{2}}{c} )^{k−j} + (ln \frac{a^{2}}{c} )^{k−j}] E_{j}(a, b, c)) \frac{t^{k}}{k!} .

(3.8)

Equating coefficients of t^{k} in (3.8) gives us

∑_{j=0}^{k} \binom{k}{j} [(ln \frac{b^{2}}{c} )^{k−j} + (ln \frac{a^{2}}{c} )^{k−j}] E_{j}(a, b, c) = 0.  (3.9)

Formula (3.5) follows.

The rest formulae follow from Definition 2.1 and formula (3.2). □

Remark 3.1. For positive numbers a, b, and c, we have the following

E_{0}(a, b, c) = 1,
E_{1}(a, b, c) = ln c − ln a − ln b,
E_{2}(a, b, c) = (ln c − 2 ln a)(ln c − 2 ln b),
E_{3}(a, b, c) = [(ln c − ln a − ln b)^{2} − 3(ln b − ln a)^{2}](ln c − ln a − ln b).

(3.10)

4 Q.-M. LUO AND F. QI

4. Some properties of the generalized Euler polynomials

In this section, we will investigate properties of the generalized Euler polynomials defined by Definition 2.2. Theorem 4.1. For any given positive numbers a, b, c and x ∈ ℝ, we have

E_{k}(x; a, b, c) = ∑_{j=0}^{k} \binom{k}{j}\frac{(ln c)^{k−j}}{2^{j}} (x − \frac{1}{2})^{k−j} E_{j}(a, b, c),  (4.1)
E_{k}(x; a, b, c) = ∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j} (ln \frac{b}{a})^{j} (x − \frac{1}{2})^{k−j} E_{j} (\frac{ln c − 2 ln a}{2(ln b − ln a)}) , (4.2)
E_{k}(x; a, b, c) = ∑_{j=0}^{k} ∑_{ℓ=0}^{j} \binom{k}{j}\binom{j}{ℓ}\frac{(ln c)^{k−j}}{2^{j}} [ln \frac{b}{a}]^{ℓ} [ln \frac{c}{ab}]^{j−ℓ} [x − \frac{1}{2}]^{k−j} E_{ℓ}, (4.3)
E_{k}(a, b, c) = 2^{k}E_{k} (\frac{1}{2}; a, b, c) ,  (4.4)
E_{k}(x) = E_{k}(x; 1, e, e).  (4.5)

Proof. By Definition 2.1 and Definition 2.2, we have

\frac{2c^{2xt}}{b^{2t} + a^{2t}} = ∑_{k=0}^{∞} 2^{k}E_{k}(x; a, b, c)\frac{t^{k}}{k!}  (4.6)

and

\begin{aligned}\frac{2c^{2xt}}{b^{2t} + a^{2t}} & = \frac{2c^{t}}{b^{2t} + a^{2t}} · c^{(2x−1)t} \\ & = (∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(a, b, c)) (∑_{k=0}^{∞} \frac{t^{k}}{k!} (2x − 1)^{k}(ln c)^{k}) \\ & = ∑_{k=0}^{∞} (∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j}(2x − 1)^{k−j}E_{j}(a, b, c)) \frac{t^{k}}{k!} . \end{aligned}  (4.7)

Equating the coefficients of \frac{t^{k}}{k!} in (4.6) and (4.7) yields

2^{k}E_{k}(x; a, b, c) = ∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j}(2x − 1)^{k−j}E_{j}(a, b, c).

Formula (4.1) follows.

The rest formulae follow directly from substituting formulae (3.2) and (3.3) into (4.1) and taking x = \frac{1}{2} in (4.1), respectively. □

Theorem 4.2. For positive integer 1 ≤ p ≤ k, we have

\frac{∂^{p}}{∂x^{p}} E_{k}(x; a, b, c) = \frac{k!}{(k − p)!}(ln c)^{p}E_{k−p}(x; a, b, c),  (4.8)
∫_{β}^{x} E_{k}(t; a, b, c)dt = \frac{1}{(k + 1) ln c} [E_{k+1}(x; a, b, c) − E_{k+1}(β; a, b, c)] .  (4.9)

Proof. Differentiating on both sides of formula (4.1) yields

\frac{∂}{∂x}E_{k}(x; a, b, c) = k(ln c)E_{k−1}(x; a, b, c).  (4.10)

Using formula (4.10) and by mathematical induction, the formulae (4.8) follows. Rearranging formula (4.10) produces

E_{k}(x; a, b, c) = \frac{1}{(k + 1) ln c} \frac{∂}{∂x}E_{k+1}(x; a, b, c).  (4.11)

Formula (4.9) follows from integrating on both sides of formula (4.11). □

Theorem 4.3. For positive numbers a, b, c and x ∈ ℝ, we have

E_{k}(x + 1; a, b, c) = ∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j}E_{j}(x; a, b, c),  (4.12)
E_{k}(x + 1; a, b, c) = 2x^{k}(ln c)^{k}
+ ∑_{j=0}^{k} \binom{k}{j} [(ln c)^{k−j} − (ln b)^{k−j} − (ln a)^{k−j}] E_{j}(x; a, b, c),

(4.13)

E_{k}(x + 1; a, b, c) = E_{k} (x; \frac{a}{c}, \frac{b}{c}, c) .  (4.14)

Proof. From Definition 2.2 and straightforward calculation, we have

\begin{aligned}\frac{2c^{xt}}{b^{t} + a^{t}} · c^{t} & = [∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(x; a, b, c)] [∑_{k=0}^{∞} \frac{t^{k}}{k!} (ln c)^{k}] \\ & = ∑_{k=0}^{∞} [∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j}E_{j}(x; a, b, c)] \frac{t^{k}}{k!} \end{aligned}  (4.15)

and

\frac{2c^{xt}}{b^{t} + a^{t}} · c^{t} = \frac{2c^{(x+1)t}}{b^{t} + a^{t}} = ∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(x + 1; a, b, c).  (4.16)

Therefore, from equating the coefficients of \frac{t^{k}}{k!} in (4.15) and (4.16), formula (4.12) follows.

6 Q.-M. LUO AND F. QI

Similarly, we obtain

\frac{2c^{(x+1)t}}{b^{t} + a^{t}} = ∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(x + 1; a, b, c) = 2c^{xt} + \frac{2c^{xt}}{b^{t} + a^{t}} (c^{t} − b^{t} − a^{t})

= 2 ∑_{k=0}^{∞} \frac{t^{k}}{k!} x^{k}(ln c)^{k} + [∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(x; a, b, c)] [∑_{k=0}^{∞} ((ln c)^{k} − (ln b)^{k} − (ln a)^{k}) \frac{t^{k}}{k!} ]

= ∑_{k=0}^{∞} [2x^{k}(ln c)^{k} + ∑_{j=0}^{k} \binom{k}{j} [(ln c)^{k−j} − (ln b)^{k−j} − (ln a)^{k−j}] E_{j}(x; a, b, c)] \frac{t^{k}}{k!} .

By equating coefficients of \frac{t^{k}}{k!} , we obain formula (4.13).

Since

∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(x + 1; a, b, c) = \frac{2c^{(x+1)t}}{b^{t} + a^{t}}

= \frac{2c^{xt}}{(\frac{b}{c})^{t} + (\frac{a}{c})^{t}} = ∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k} (x; \frac{a}{c}, \frac{b}{c}, c) , (4.17)

By equating coefficients, we obtain formula (4.14). The proof is complete. □

Corollary 4.3.1. The following formulae are valid for positive numbers a, b, c and real number x:

E_{k}(x + 1) + E_{k}(x) = 2x^{k},  (4.18)
E_{k}(x + 1) = ∑_{j=0}^{k} \binom{k}{j}E_{j}(x),  (4.19)
E_{k}(x + 1; 1, b, b) + E_{k}(x; 1, b, b) = 2x^{k}(ln b)^{k},  (4.20)
E_{k}(x + 1; 1, b, b) = ∑_{j=0}^{k} \binom{k}{j}E_{j}(x; 1, b, b)(ln b)^{k−j},  (4.21)
∑_{j=0}^{k−1} \binom{k}{j}E_{j}(x; 1, b, b)(ln b)^{k−j} + 2E_{k}(x; 1, b, b) = 2x^{k}(ln b)^{k},  (4.22)
∫_{x}^{x+1} E_{k}(t; a, b, c)dt = \frac{1}{(k + 1) ln c} ∑_{j=0}^{k} \binom{k + 1}{j}(ln c)^{k−j}E_{j}(x; a, b, c). (4.23)

Theorem 4.4. For positive numbers a, b, c > 0, x ∈ ℝ, and nonnegative integer k, we have

E_{k}(1 − x; a, b, c) = (−1)^{k}E_{k} (x; \frac{c}{a}, \frac{c}{b}, c) ,  (4.24)
E_{k}(1 − x; a, b, c) = E_{k} (−x; \frac{a}{c}, \frac{b}{c}, c) .  (4.25)

Proof. From Definition 2.2 and easy computation, we have

∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(1 − x; a, b, c) = \frac{2c^{(1−x)t}}{b^{t} + a^{t}} = \frac{2c^{t} · c^{−xt}}{b^{t} + a^{t}}

= \frac{2c^{−xt}}{(\frac{c}{b})^{−t}+(\frac{c}{a})^{−t}} = ∑_{k=0}^{∞} \frac{t^{k}}{k!} (−1)^{k}E_{k} (x; \frac{c}{a}, \frac{c}{b}, c) , (4.26)

equating coefficients of t^{k} above leads to formula (4.24).

By the same procedure, we can establish formula (4.25). □

Theorem 4.5. For positive numbers a, b, c > 0, nonnegative natural number k, and x, y ∈ ℝ, we have

E_{k}(x + y; a, b, c) = ∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j}y^{k−j}E_{j}(x; a, b, c),  (4.27)
E_{k}(x + y; a, b, c) = ∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j}x^{k−j}E_{j}(y; a, b, c).  (4.28)

Proof. These two formulae can be deduced from the following calculation and considering symmetry of x and y:

∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(x + y; a, b, c) = \frac{2c^{(x+y)t}}{b^{t} + a^{t}} = \frac{2c^{xt} · c^{yt}}{b^{t} + a^{t}}

= [∑_{k=0}^{∞} \frac{t^{k}}{k!} E_{k}(x; a, b, c)] [∑_{k=0}^{∞} \frac{t^{k}}{k!} (ln c)^{k}y^{k}]

= ∑_{k=0}^{∞} [∑_{j=0}^{k} \binom{k}{j}(ln c)^{k−j}y^{k−j}E_{j}(x; a, b, c)] \frac{t^{k}}{k!} . (4.29)

The proof is complete. □

Theorem 4.6. For natural numbers k and m, and positive number b, we have

∑_{ℓ=1}^{m}(−1)^{ℓ}ℓ^{k} = \frac{1}{2(ln b)^{k}} [(−1)^{m}E_{k}(m + 1; 1, b, b) − E_{k}(1; 1, b, b)] .  (4.30)

Proof. Rearraging formula (4.20) give us

x^{k} = \frac{1}{2(ln b)^{k}} [E_{k}(x + 1; 1, b, b) + E_{k}(x; 1, b, b)] .  (4.31)

Replacing x by ℓ ∈ ℕ and suming up ℓ from 1 to m yields

\begin{aligned}∑_{ℓ=1}^{m}(−1)^{ℓ}ℓ^{k} & = \frac{1}{2(ln b)^{k}} ∑_{ℓ=1}^{m}(−1)^{ℓ} [E_{k}(ℓ + 1; 1, b, b) + E_{k}(ℓ; 1, b, b)] \\ & = \frac{1}{2(ln b)^{k}} [(−1)^{m}E_{k}(m + 1; 1, b, b) − E_{k}(1; 1, b, b)] . \end{aligned}  (4.32)

The proof is complete. □

Remark 4.1. In the final, we give several concrete formulae as follows:

E_{0}(x; a, b, c) = 1,  (4.33)
E_{1}(x; a, b, c) = x ln c − \frac{1}{2}(ln a + ln b),  (4.34)
E_{2}(x; a, b, c) = (x − \frac{1}{2})^{2} (ln c)^{2} + (x − \frac{1}{2}) (ln c − ln b − ln a) ln c  (4.35)
+ \frac{1}{4}(ln c − 2 ln a)(ln c − 2 ln b).  (4.36)

References

[1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tebles, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, Washington, 1965.

[2] B.-N. Guo and F. Qi, Generalisation of Bernoulli polynomials, Internat. J. Math. Ed. Sci. Tech. 33 (2002), no. 3, in press. RGMIA Res. Rep. Coll. 4 (2001), no. 4, Art. 10, 691–695.

Available online at http://rgmia.vu.edu.au/v4n4.html.

[3] Q.-M. Luo, B.-N. Guo, and F. Qi, Generalizations of Bernoulli’s numbers and polynomials, RGMIA Res. Rep. Coll. 5 (2001), no. 2. Available online at http://rgmia.vu.edu.au/v5n2.

html.

[4] Zh.-X. Wang and D.-R. Guo, Tèshū Hánshù Gàilùn (Introduction to Special Function), The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000. (Chinese)

(Luo) Department of Broadcast-Television-Teaching, Jiaozuo University, Jiaozuo City, Henan 454002, China

(Qi) Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Jiaozuo City, Henan 454000, China

E-mail address: qifeng@jzit.edu.cn or qifeng618@hotmail.com
URL: http://rgmia.vu.edu.au/qi.html