R E S E A R C H
Open Access
Some convolution identities for
Frobenius-Euler polynomials
Jing Pan and Fengzao Yang
**Correspondence:
[email protected] Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, People’s Republic of China
Abstract
In this paper, by applying the generating function methods and summation transform techniques, we establish some new convolution identities for the Frobenius-Euler polynomials. It turns out that some well-known results are obtained as special cases.
MSC: 11B68; 11S40; 05A19
Keywords: Frobenius-Euler polynomials and numbers; Euler polynomials and numbers; Bernoulli polynomials and numbers; combinatorial identities
1 Introduction
The classical Frobenius-Euler polynomialsHn(x|λ) are usually defined by the generating function:
–λ
et–λe xt=
∞
n= Hn(x|λ)
tn n!
|t|<πifλ= –;|t|<log(/λ) otherwise. (.)
In particular, the casex= in (.) is called the classical Frobenius-Euler numbers given byHn(λ) =Hn(|λ). It is worthy of mentioning that the classical Frobenius-Euler poly-nomials and numbers was firstly introduced and studied in great detail by Frobenius []. We also refer to [–] for some interesting properties on the classical Frobenius-Euler polynomials and numbers.
The widely investigated analogs of the classical Frobenius-Euler polynomials are the classical Bernoulli polynomialsBn(x) and the classical Euler polynomialsEn(x), which are usually defined by the generating functions (see,e.g., [–]):
text et– =
∞
n= Bn(x)
tn
n! (|t|< π),
ext et+ =
∞
n= En(x)
tn n!
|t|<π.
(.)
The rational numbersBnand integersEngiven by
Bn=Bn(), En= nEn(/), (.)
are called the classical Bernoulli numbers and the classical Euler numbers, respectively. Obviously, the caseλ= – in (.) gives the classical Euler polynomials. In fact, the classical Bernoulli polynomials can also be expressed by the classical Frobenius-Euler numbers, as follows:
mn– m–
i=
λiBn
i m
= n
λ– Hn–
λ
(m,n≥;λ= ). (.)
In the year , Agoh and Dilcher [] made use of some connections between the classical Bernoulli numbers and the Stirling numbers of the second kind to extend Euler’s well-known recurrence formula on the classical Bernoulli numbers:
n
i=
n i
BiBn–i= –nBn–– (n– )Bn (n≥), (.)
and they obtained a convolution identity on the classical Bernoulli numbers, as follows:
n
i=
n i
Bm+n–iBk+i
= –k!·m!·(n+δ(k,m)(k+m+ )) (k+m+ )! Bk+m+n
+ k+m
i=
(–)i Bk+m+–i k+m+ –i
(–)k
k+ i
k+ –i k+ n–
i k+ m
+ (–)m
m+ i
m+ –i m+ n–
i m+ k
Bn+i–, (.)
wherek,m,nare non-negative integers,δ(k,m) = whenk= orm= , andδ(k,m) = otherwise. Interest in (.) stems from its good value distributions, some authors reproved the above formula by applying different methods; see, for example [–].
Motivated and inspired by the work of the above authors, in this paper we establish some similar convolution identities for the classical Frobenius-Euler polynomials to (.) by ap-plying the generating function methods and summation transform techniques developed in []. Accordingly we present some special cases as well as immediate consequences of the main results.
This paper is organized as follows. In the second section, we state some new convolution identities for the classical Frobenius-Euler polynomials, by virtue of which some known results including the classical ones due to Carlitz [] are obtained as special cases. In the third and fourth sections are contributions to the proofs of the new convolution identities for the classical Frobenius-Euler polynomials.
2 The statement of results
Theorem . Let m,n,k be non-negative integers.Then,forλ= , ,μ= , andλμ= ,
It follows that we give some special cases of Theorem .. Since the classical Frobenius-Euler polynomials obey the symmetric distributions (see,e.g., [])
Since the classical Frobenius-Euler polynomials satisfy the difference equation (see,e.g., [])
Hn( +x|λ) –λHn(x|λ) = ( –λ)xn (n≥;λ= ), (.)
from (.) and (.), we obtain
(–)nHn
x
λ
= ( –λ)(–x)n+λHn(–x|λ) (n≥;λ= , ). (.)
Hence, by applying (.) to (.), we get the following formula for the products of the classical Frobenius-Euler polynomials.
Corollary . Let m,n be non-negative integers.Then,forλ= ,μ= ,andλμ= ,
Hm(x|λ)Hn(y|μ)
=λ(μ– )
λμ– m
i=
m i
Hm–i(x–y|λ)Hn+i(y|λμ)
+μ(λ– )
λμ– n
i=
n i
Hn–i(y–x|μ)Hm+i(x|λμ)
–(λ– )(μ– )
λμ– m
i=
m i
(x–y)m–iHn+i(y|λμ). (.)
In particular, the case x=yin Corollary . gives for non-negative integersm,nand
λ= ,μ= ,λμ= ,
Hm(x|λ)Hn(x|μ)
=λ(μ– )
λμ– m
i=
m i
Hi(λ)Hm+n–i(x|λμ)
+μ(λ– )
λμ– n
i=
n i
Hi(μ)Hm+n–i(x|λμ)
–(λ– )(μ– )
λμ– Hm+n(x|λμ), (.)
which was firstly discovered by Carlitz []. Corollary . can also be found in [], The-orem ., where it was used to give, for non-negative integernandλ=±,
n
i=
Hi(x|λ)Hn–i(x|λ)
= λ
λ+ n
i=
n+ i
Hn–i(λ)Hi
x|λ–λ–
λ+ (n+ )Hn
and for positive integern≥ andλ=±,
n–
i=
Hi(x|λ)Hn–i(x|λ) i(n–i)
= λ
λ+ · n
n–
i=
n i
(Hn––Hi–)Hi(λ)Hn–i
x|λ+ Hn–
Hn(x|λ)
n , (.)
whereHnis the Harmonic numbers given by
H= and Hn= n
i= i = +
+· · ·+
n (n≥). (.)
Equations (.) and (.) are very analogous to the following convolution identities on the classical Bernoulli polynomials due to Kimet al.[], namely
n
i=
Bi(x)Bn–i(x) = n+
n–
i=
n+ i
Bn–iBi(x) + (n+ )Bn(x) (n≥), (.)
and
n–
i=
Bi(x)Bn–i(x) i(n–i) =
n
n–
i=
n i
Bn–iBi(x) n–i +
nHn–Bn(x) (n≥). (.)
The casex= in (.) and (.) will lead to the famous Miki identity and the famous Matiyasevich identity on the classical Bernoulli numbers, respectively. For some related results of (.), (.), (.), and (.), one may consult [, –].
Theorem . Let m,n,k be non-negative integers.Then,forλ= , ,
n
i=
n i
βm+n–i(x)Hk+i(y|λ)
=
λ– k
i=
k i
(–)k–iHm+k–i
x
λ
Hn+i(x+y|λ)
+ m
i=
m i
(–)m–iHm+k–i(y|λ)βn+i(x+y)
– (–)m m!·n!
(m+n+ )!Hm+n+k+(y|λ), (.)
whereβn(x)is denoted byβn(x) =Bn+(x)/(n+ )for non-negative integer n.
It is clear that the classical Bernoulli polynomials satisfy the symmetric distributions (see,e.g., []):
which means
βn( –x) = (–)n+βn(x) (n≥). (.)
Hence, by setting x+y= –zin Theorem ., in view of (.) and (.), we get the following result.
Corollary . Let m,n,k be non-negative integers.Then,forλ= , and x+y+z= ,
(–)n n
i=
n i
βm+n–i(x)Hk+i(y|λ) + (–)k –λ
k
i=
k i
Hm+k–i
x
λ
Hn+i
z
λ
+ (–)m m
i=
m i
Hm+k–i(y|λ)βn+i(z)
= (–)m+n+ m!·n!
(m+n+ )!Hm+n+k+(y|λ), (.)
whereβn(x)is denoted byβn(x) =Bn+(x)/(n+ )for non-negative integer n.
If we takeλ= – in Corollary ., we obtain the following convolution identity for the classical Euler polynomials.
Corollary . Let m,n,k be non-negative integers.Then,for x+y+z= ,
(–)n n
i=
n i
βm+n–i(x)Ek+i(y) + (–)k
k
i=
k i
Em+k–i(x)En+i(z)
+ (–)m m
i=
m i
Em+k–i(y)βn+i(z)
= (–)m+n+ m!·n!
(m+n+ )!Em+n+k+(y), (.)
whereβn(x)is denoted byβn(x) =Bn+(x)/(n+ )for non-negative integer n.
If we substitutey–xforyin Theorem ., we get for non-negative integersm,n,kand
λ= , ,
n
i=
n i
βm+n–i(x)Hk+i(y–x|λ)
=
λ– k
i=
k i
(–)k–iHm+k–i
x
λ
Hn+i(y|λ)
+ m
i=
m i
(–)m–iHm+k–i(y–x|λ)βn+i(y)
– (–)m m!·n!
By settingm= in (.), we get for non-negative integersn,kandλ= , ,
βn(y)Hk(y–x|λ)
= n
i=
n i
βn–i(x)Hk+i(y–x|λ)
–
λ– k
i=
k i
(–)k–iHk–i
x
λ
Hn+i(y|λ) +
n+ Hn+k+(y–x|λ). (.)
Substitutingxfory,x–yforx,mforn, andnforkin (.) gives
βm(x)Hn(y|λ)
= m
i=
m i
βm–i(x–y)Hn+i(y|λ)
–
λ– n
i=
n i
(–)n–iHn–i
x–y
λ
Hm+i(x|λ) +
m+ Hm+n+(y|λ). (.)
It follows from (.) and (.) that, for non-negative integersm,nandλ= ,
Bm+(x)Hn(y|λ) = m+
i=
m+ i
Bm+–i(x–y)Hn+i(y|λ)
– λ
λ– (m+ ) n
i=
n i
Hn–i(y–x|λ)Hm+i(x|λ)
+ (m+ ) n
i=
n i
(y–x)n–iHm+i(x|λ). (.)
Notice that, for non-negative integersm,n(see,e.g., [], Theorem .),
m
i=
m i
xm–ifn+i(y) = n
i=
n i
(–x)n–ifm+i(x+y), (.)
wherefn(x) is a sequence of polynomials given for formal power seriesF(t) by
∞
n= fn(x)
tn n!=F(t)e
(x–/)t. (.)
If we takeF(t) = ( –λ)et//(et–λ) in (.) and then substitutex–yforxin (.), we obtain for non-negative integersm,nandλ= ,
m
i=
m i
(x–y)m–iHn+i(y|λ) = n
i=
n i
(y–x)n–iHm+i(x|λ). (.)
Corollary . Let m,n non-negative integers.Then,forλ= ,
Bm+(x)Hn(y|λ) = m+
i=
m+ i
Bm+–i(x–y)Hn+i(y|λ)
– λ
λ– (m+ ) n
i=
n i
Hn–i(y–x|λ)Hm+i(x|λ)
+ (m+ ) m
i=
m i
(x–y)m–iHn+i(y|λ). (.)
If we takek= in (.) then, for non-negative integersm,nandλ= , ,
Hm
x
λ
Hn(y|λ) = –(λ– ) m
i=
m i
(–)m–iHm–i(y–x|λ)βn+i(y)
+ (λ– ) n
i=
n i
βm+i(x)Hn–i(y–x|λ)
+ (λ– )(–) mm!·n!
(m+n+ )!Hm+n+(y–x|λ), (.)
which together with (.) yields the following result.
Corollary . Let m,n be non-negative integers.Then,λ= , ,
Hm
x
λ
Hn(y|λ)
=
λ–
m
i=
m i
Hm–i
x–y
λ
Bn+i+(y) n+i+
+ (λ– ) n
i=
n i
Hn–i(y–x|λ)
Bm+i+(x) m+i+
–(λ– )
λ
m
i=
m i
(x–y)m–iBn+i+(y) n+i+
+ (λ– )(–) mm!·n!
(m+n+ )!Hm+n+(y–x|λ). (.)
In particular, the casex=yin Corollary . gives that ifλ= then, for positive integer mand non-negative integern,
Bm(x)Hn(x|λ)
= m
i=
m i
BiHm+n–i(x|λ)
– λ
λ– m n
i=
n i
and if we setx=yand substitutemfornandnformin Corollary . then, for positive integersm,nandλ= , ,
Hm(x|λ)Hn
x
λ
= (λ– ) m
i=
m i
Hi(λ)
Bm+n+–i(x) m+n+ –i+
λ–
n
i=
n i
Hi
λ
Bm+n+–i(x) m+n+ –i
+ (–)n(λ– ) m!·n!
(m+n+ )!Hm+n+(λ). (.)
Equations (.) and (.) were firstly discovered by Carlitz [] who used them to give the expressions of the products of the classical Bernoulli polynomials and the classical Euler polynomials stated in Nielsen’s classical book []. For different proofs of Corollar-ies . and ., see [] for details. For some related results on the products of the classical Bernoulli and Euler polynomials, one can refer to [–].
3 The proof of Theorem 2.1
We first prove the following auxiliary result.
Lemma . Let m,n,k be non-negative integers.Then,forλ= , ,μ= , ,andλμ= ,
k
j=
k j
n
i=
n i
Hi+j
x
μ
Hm+n+k–i–j
y
λμ
=λμ–
λ– Hm
y
λ
Hn+k
x+y
μ
–λ(μ– )
λ– m
i=
m i
(–)m–iH
m–i(x|λ)Hn+k+i
x+y
λμ
. (.)
Proof It is easily seen that
λeu– ·
μev– =
λeu
λeu– +
μev–
λμeu+v– . (.)
If we multiply both sides of the above identity by (μ– )(λμ– )exv+y(u+v), we get
μ–
μev– e
xv· λμ–
λμeu+v– e y(u+v)
=λμ–
λ– ·
λ–
λeu– e
yu· μ–
μev– e (x+y)v
–λ(μ– )
λ– ·
λ–
λeu– e
(–x)u· λμ–
λμeu+v– e
(x+y)(u+v). (.)
By substituting /λforλin (.), we have
λ–
λeu– e xu=
∞
m= Hm
x
λ
um
More generally, by the Taylor theorem, we discover
If we takektimes the derivative for (.) with respect tov, in view of the Leibniz rule, we obtain
which together with the Cauchy product and (.) yields
∞
We now give the detailed proof of Theorem ..
The-orem . holds for all positive integers less than k. It follows from Lemma . that, for non-negative integersm,nand positive integerk,
n
Since Theorem . holds for all positive integers less thank, we have
k–
Notice that, for non-negative integersm,n,k,i,
k
by using induction onk. It follows from (.) that, for non-negative integersm,n,k,iwith k≥,
Thus, by applying (.) and (.) to (.) and combining with (.) we have the desired
4 The proof of Theorem 2.4
In a similar consideration to Theorem ., we firstly give the following result.
Lemma . Let m,n,k be non-negative integers.Then,forλ= , ,
which together with the Taylor theorem gives
ex(u+v)
whereMis given by
Observe that
which together with the binomial theorem means
∞
By changing the order of the summation in the right side of (.), we have
M= (λ– )
Notice that, for non-negative integerm,nwithm≥n+ ,
m–
by using induction onm. Hence, applying (.) to (.) gives
M= (λ– )
By putting (.) to (.) and then takingktimes the derivative with respect tov, we get
Thus, replacingybyx+yand comparing the coefficients ofum·vn/m!·n! in (.) gives
Lemma ..
We next give the detailed proof of Theorem ..
Proof of Theorem. Obviously, Theorem . holds trivially whenk= in Lemma .. Now, we assume Theorem . holds for all positive integers less thank. It follows from Lemma . that, for non-negative integersm,nand positive integerk,
n
i=
n i
βm+n–i(x)Hk+i(y|λ)
=
λ– Hm
x
λ
Hn+k(x+y|λ)
+ m+k
i=
m i–k
(–)m+k–iHm+k–i(y|λ)βn+i(x+y)
– (–)m m!·(n+k)!
(m+n+k+ )!Hm+n+k+(y|λ)
– k–
j=
k j
n
i=
n i
Hi+j(y|λ)βm+n+k–i–j(x). (.)
Since Theorem . holds for all positive integers less thank, we get
k–
j=
k j
n
i=
n i
Hi+j(y|λ)βm+n+k–i–j(x)
=
λ– k–
i= (–)iH
n+i(x+y|λ)Hm+k–i
x
λ
k–
j= (–)j
k j
j i
+ m+k
i=
(–)m+k–iHm+k–i(y|λ)βn+i(x+y) k–
j= (–)j
k j
m+k–j i
–Hm+n+k+(y|λ) n+
k–
j=
(–)m+k–j
k
j
m+n+k+–j
n+
. (.)
Observe that, for non-negative integersm,n,k,
k
j= (–)j
k
j
m+n+j
n
=m!·(n+k– )!·n
(m+n+k)! , (.)
by using induction onk. It follows from (.) that, for non-negative integers m,nand positive integerk,
n+
k–
j=
(–)m+k–j
k j
m+n+k+–j
n+
=(–)
mm!·(n+k)!
(m+n+k+ )! –
(–)mm!·n!
Thus, by applying (.), (.), and (.) to (.) and combining with (.), we get the
desired result. This concludes the proof of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors express their gratitude to the anonymous referees for their helpful comments and suggestions in improving this paper. This work is supported by the Foundation for Fostering Talents in Kunming University of Science and Technology (Grant No. KKSY201452090) and the National Natural Science Foundation of P.R. China (Grant No. 51406071, 51666006, 61305057).
Received: 2 September 2016 Accepted: 3 December 2016
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