Journal of Number Theory 133 (2013) 822–824

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Journal of Number Theory

www.elsevier.com/locate/jnt

Corrigendum

Corrigendum to “Identities involving Frobenius–Eulerpolynomials arising from non-linear differential equations”[J. Number Theory 132 (12) (2012) 2854–2865]

Taekyun Kim

Department Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

a r t i c l e i n f o

Article history:Received 8 August 2012Accepted 29 August 2012Available online 15 September 2012Communicated by David Goss

Keywords:Euler numbersFrobenius–Euler numbers and polynomialsSums product of Euler numbersDifferential equations

Our paper may contain some typographical errors. The changes are as follows:1. Our Eq. (7) in p. 2856 is typographical error. The equation must be replaced 1

et−u by 1u−et .

We rewritten Eq. (7) in p. 2856.

F = F (t) = 1

u − et, and F N(t, x) = F × · · · × F︸ ︷︷ ︸

N-times

ext for N ∈ N.

2. In Theorem 1 and Corollary 2, 1et−u must be changed by F (t) = 1

u−et .Therefore, we rewritten Theorem 1 and Corollary 2 as follows:

DOI of original article: http://dx.doi.org/10.1016/j.jnt.2012.05.033.E-mail address: tkkim@kw.ac.kr.

0022-314X/$ – see front matter © 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jnt.2012.08.002

T. Kim / Journal of Number Theory 133 (2013) 822–824 823

Theorem 1. For u ∈ C with u �= 1, and N ∈ N, let us consider the following non-linear differential equationwith respect to t:

F N(t) = N

uN−1

N−1∑k=0

1

(k + 1)!∑

l1+···+lk+1=N

1

l1l2 · · · lk+1F (k)(t), (52)

where F (k)(t) = dk F (t)dtk and F N (t) = F (t) × · · · × F (t)︸ ︷︷ ︸

N-times

. Then F (t) = 1u−et is a solution of (52).

Corollary 2. For N ∈N, we set

F N(t, x) = N

uN−1

N∑k=0

1

(k + 1)!∑

l1+···+lk+1=N

1

l1l2 · · · lk+1F (k)(t, x). (53)

Then etx

u−et is a solution of (53).

3. P. 2862, line 7, line 8, and line 9 from bottom should be replaced (1−u) by (u−1). We rewrittenthose equations as follows:

F N(t) =(

1

u − et

)×

(1

u − et

)× · · · ×

(1

u − et

)︸ ︷︷ ︸

N-times

= 1

(u − 1)N

(1 − u

et − u

)×

(1 − u

et − u

)× · · · ×

(1 − u

et − u

)︸ ︷︷ ︸

N-times

= 1

(u − 1)N

∞∑l=0

H (N)

l (u)tl

l! , (55)

and

F (t) =(

1 − u

et − u

)(1

u − 1

)= 1

u − 1

∞∑l=0

Hl(u)tl

l! .

From (55), we note that

F (k)(t) = dk F (t)

dtk= 1

u − 1

∞∑l=0

Hl+k(u)tl

l! . (56)

4. Theorem 3, Corollary 4 and Corollary 5 should be replaced (1 − u) by (u − 1). We rewrittenTheorem 3, Corollary 4 and Corollary 5.

Theorem 3. For N ∈ N, n ∈ Z+ , we have

H (N)n (u) = N

(u − 1

u

)N−1 N−1∑k=0

1

(k + 1)!∑

l +···+l =N

Hn+k(u)

l1l2 · · · lk+1.

1 k+1

824 T. Kim / Journal of Number Theory 133 (2013) 822–824

Corollary 4. For N ∈N, n ∈ Z+ , we have

∑l1+···+lN=n

(n

l1, . . . , lN !)

Hl1(u)Hl2(u) · · · HlN (u)

= N

(u − 1

u

)N−1 N−1∑k=0

1

(k + 1)!∑

l1+···+lk+1=N

Hn+k(u)

l1l2 · · · lk+1.

Corollary 5. For N ∈N, n ∈ Z+ , we have

H (N)n (x|u)

= N

(u − 1

u

)N−1 N−1∑k=0

1

(k + 1)!∑

l1+···+lk+1=N

1

l1l2 · · · lk+1

n∑m=0

(n

m

)Hm+k(u)xn−m.