A chi um Ma hema icum
Luis M. Na as; F ancisco J. Ruiz; Juan L. Va ona
Exis ence and educ ion o gene alized Apos ol-Be noulli, Apos ol-Eule and
Apos ol-Genocchi polynomials
A chi um Ma hema icum, Vol. 55 (2019), No. 3, 157–165
Pe sis en URL: h p://dml.cz/dmlcz/147823
Te ms o use:
© Masa yk Uni e si y, 2019
Ins i u e o Ma hema ics o he Czech Academy o Sciences p o ides access o digi ized documen s
s ic ly o pe sonal use. Each copy o any pa o his documen mus con ain hese Te ms o use.
This documen has been digi ized, op imized o elec onic deli e y and
s amped wi h digi al signa u e wi hin he p ojec DML-CZ: The Czech Digi al
Ma hema ics Lib a y h p://dml.cz
ARCHIVUM MATHEMATICUM (BRNO)
Tomus 55 (2019), 157–165
EXISTENCE AND REDUCTION OF GENERALIZED
APOSTOL-BERNOULLI, APOSTOL-EULER AND
APOSTOL-GENOCCHI POLYNOMIALS
Luis M. Na as, F ancisco J. Ruiz, and Juan L. Va ona
Abs ac . One can ind in he ma hema ical li e a u e many ecen pape s
s udying he gene alized Apos ol-Be noulli, Apos ol-Eule and Apos ol-Genocchi
polynomials, de ined by means o gene a ing unc ions. In his a icle we cla i y
he ange o pa ame e s in which hese de ini ions a e alid and when hey
p o ide essen ially di e en amilies o polynomials. In pa icula , we show
ha , up o mul iplica i e cons an s, i is enough o ake as he “main amily”
hose gi en by
2
λe + 1 α
ex =
∞
X
n=0
E(α)
n(x;λ) n
n!, λ ∈C {−1},
and as an “excep ional amily”

e −1α
ex =
∞
X
n=0
B(α)
n(x) n
n!,
bo h o hese o α∈C.
1. In oduc ion
The gene alized Apos ol-Be noulli, Apos ol-Eule and Apos ol-Genocchi polyno-
mials o o de
α∈N∪ {
0
}
a e de ined, espec i ely, by means o he gene a ing
unc ions and se ies expansions

λe −1αex =
∞
X
n=0
B(α)
n(x;λ) n
n!,(1)
2
λe + 1αex =
∞
X
n=0
E(α)
n(x;λ) n
n!,(2)
2010 Ma hema ics Subjec Classi ica ion: p ima y 11B68; seconda y 05A15.
Key wo ds and ph ases: Be noulli polynomials, Nø lund polynomials, Apos ol-Be noulli poly-
nomials, Apos ol-Eule polynomials, Apos ol-Genocchi polynomials, gene a ing unc ions, Appell
sequences.
The au ho s a e suppo ed by g an MTM2015-65888-C4-4-P o he Minis e io de Economía
y Compe i i idad (Spain). The second au ho has also been suppo ed by P ojec E-64, D. G.
A agón (Spain).
Recei ed Sep embe 8, 2018. Edi o M. Kolář.
DOI: 10.5817/AM2019-3-157
158 L.M. NAVAS, F.J. RUIZ AND J.L. VARONA
2
λe + 1αex =
∞
X
n=0
G(α)
n(x;λ) n
n!.(3)
These a e alid in a sui able neigbou hood o
= 0, whe e
λ
is (wi h some excep ions)
any complex numbe . They a e gene aliza ions o he classical Be noulli, Eule and
Genocchi polynomials
Bn
(
x
),
En
(
x
)and
Gn
(
x
), ha co espond o he cases
λ
= 1
and
α
= 1 (mo eo e , he so-called Be noulli, Eule and Genocchi numbe s a e
Bn=Bn(0),En= 2nEn(1
2)and Gn=Gn(0)).
In he ma hema ical li e a u e, he pa ame e s
α
and
λ
ha e been included inde-
penden ly (we gi e some his o ical de ails in Subsec ion 1.1); once he pa ame e s
ha e been used oge he , he de ini ions
(1)
,
(2)
and
(3)
ha e been ex ended o
α∈C
. The goal o his pape is o cla i y when his ex ension is possible (Subsec-
ion 1.2), and o educe he abo e-men ioned de ini ions wi h complex
λ
and
α
o a
smalle class o polynomials ha supp esses he i ial ela ionships be ween hem
(Sec ion 2, see
(7)
,
(8)
and De ini ion 4). Excep o mul iplica i e cons an s, ou
educ ion co e s e e y case o gene alized Apos ol-Be noulli, Apos ol-Eule and
Apos ol-Genocchi polynomials wi hou he necessi y o adding ex a pa ame e s.
As usual, in his pape we will always use he p incipal b anch o complex
powe s, in pa icula 1
α
= 1 o
α∈C
(bu no hing subs an ial changes wi h a
di e en choice).
1.1.
One pa ame e .
Fo Be noulli polynomials, which a e he ones mos o en
discussed, he complex pa ame e
λ
was in oduced by Apos ol in 1951 [
1
], in connec-
ion wi h he Le ch ze a unc ion, and hus he unc ions
Bn
(
x
;
λ
) =
B(1)
n
(
x
;
λ
)a e
called he Apos ol-Be noulli polynomials; he in ege pa ame e
α
was in oduced
by Nø lund in 1922 [
10
], and hus
B(α)
n
(
x
; 1) a e also known as Nø lund polynomials.
La e in his pape we will gi e mo e de ails abou he co esponding gene aliza ions
o Eule and Genocchi polynomials, and he use o bo h he pa ame e s
λ
and
α
.
In o de o a oid con usion, i is impo an o no e ha , in he abo e de ini ions,
he
n
h polynomial is no always a polynomial o deg ee
n
( his will no longe
happen wi h he educ ion ha we gi e in De ini ion 4). Le us analyze his, as
well as some o he ele an de ails and ela ions.
Fo ixed
α
= 1, he alue
λ
= 1 co esponds o he classical Be noulli poly-
nomials, i.e.
B(1)
n
(
x
; 1) =
Bn
(
x
), bu i is ce ainly no he case ha
Bn
(
x
) =
limλ→1B(1)
n
(
x
;
λ
). The e is a limi ing ela ionship be ween
B(1)
n
(
x
;
λ
)and
Bn
(
x
)as
λ→
1, bu i is no immedia ely ob ious. Ano he aspec o his discon inui y is
ha , al hough
Bn
(
x
)is monic o deg ee
n
, o
λ6
= 1 he deg ee o
B(1)
n
(
x
;
λ
)is
n−1and i s leading e m is n/(λ−1).
The case
λ
= 0 is i ial; indeed
B(1)
0
(
x
; 0) = 0 and
B(1)
n
(
x
; 0) =
−nxn−1
o
n≥
1. Fo his eason, i is usual o assume
λ6
= 0, bu we do no need his
es ic ion in wha ollows.
Again o
α
= 1, he Apos ol-Eule and Apos ol-Genocchi polynomials do no
in oduce eal no el ies, because hey can be educed o he Apos ol-Be noulli
EXISTENCE AND REDUCTION OF POLYNOMIALS 159
amily. W i ing he gene a ing unc ion (2) as
2
λe + 1ex =−2

(−λ)e −1ex
and using he uniqueness o Taylo expansions, i is clea ha
E(1)
n(x;λ) = −2
n+ 1B(1)
n+1(x;−λ)
o all
λ∈C
. Appa en ly, his simple ela ion has no been no ed be o e [
9
],
because one inds p e ious pape s which s udy p ope ies o Apos ol-Be noulli and
Apos ol-Eule as di e en amilies o polynomials (see, o ins ance, [
6
] o [
2
]). In
he same way, o Apos ol-Genocchi polynomials we ha e
G(1)
n+1(x;λ)=(n+ 1)E(1)
n(x;λ) = −2B(1)
n+1(x;−λ)
o all
λ∈C
. As in he case o Apos ol-Be noulli polynomials, when
λ6
=
−
1 he
polynomial G(1)
n(x;λ)has deg ee n−1, and G(1)
0(x;λ)=0.
As men ioned abo e, he in oduc ion o he pa ame e
α
o Be noulli polyno-
mials (i.e., wi h
λ
= 1) is due o Nø lund in 1922 [
10
]; hey a e he so-called gene a-
lized Be noulli polynomials o o de
α
,
B(α)
n
(
x
) =
B(α)
n
(
x
; 1). Two yea s la e , in [
11
,
p. 120], Nø lund de ined he gene alized Eule polynomials
E(α)
n
(
x
) =
E(α)
n
(
x
; 1).
Mo eo e , he also s udies he case when
α
is a nega i e in ege , bo h o Eule and
o Be noulli polynomials (see [
11
, p. 130]). The gene alized Genocchi polynomials
o o de α,G(α)
n(x) = G(α)
n(x; 1), appea much la e , and can be ound in [4].
1.2.
Two pa ame e s.
The s udy o bo h pa ame e s
λ
and
α
simul aneously is so-
mewha ecen . Fu he mo e, he es ic ion
α∈N∪{
0
}
as in
(1)
,
(2)
and
(3)
has no
been included in he co esponding de ini ions. The gene alized Apos ol-Be noulli
and Apos ol-Eule polynomials o ( eal o complex) o de
α
we e de ined in 2005 by
Luo and S i as a a [
7
], and in 2011 he same au ho s in oduced and in es iga ed
he gene alized Apos ol-Genocchi polynomials o ( eal o complex) o de
α
[
8
]; hey
can also be ound in [12, §1.9, p. 91].
In he le hand sides o
(1)
,
(2)
and
(3)
we ha e unc ions
g
(
) ha ( o e e y
ixed
x
) mus be expanded in powe s o
. O cou se, a unc ion
g
(
) ha is no
analy ic in a disk a ound
= 0, canno be expanded in powe s o
, in he same
way ha , o ins ance, we canno w i e
1/2
o
−1
as
P∞
n=0 an n
. This is wha
happens wi h some o hese e y gene al assump ions. Fo λ6= 1,
λe −1≈(λ−1)−1· when →0,
and
α
is no analy ic a ound
= 0 when
α /∈N∪{
0
}
; consequen ly, he expansion
(1)
(and hence also
B(α)
n
(
x
;
λ
)) does no exis o any eal o complex
α
, only o
α∈N∪ {
0
}
. Fo
λ
= 1,
/
(
e −
1)
≈
1when
→
0, and in his case we can
indeed de ine he Nø lund polynomials
B(α)
n
(
x
; 1) o a bi a y
α∈C
. A simila
beha io occu s in he expansion
(3)
. Thus, he polynomials
G(α)
n
(
x
;
λ
)only exis
o
α∈N∪ {
0
}
when
λ6
=
−
1, and
G(α)
n
(
x
;
−
1) can be de ined o a bi a y
α∈C
.
160 L.M. NAVAS, F.J. RUIZ AND J.L. VARONA
(Bu
G(α)
n
(
x
;
−
1) = (
−
2)
αB(α)
n
(
x
; 1), so no hing essen ially new is in oduced in
his case.)
The expansion
(2)
is less es ic i e, and i does accep any
α∈C
in mos cases.
Fo λ6=−1,
2
λe + 1 ≈2(λ+ 1)−16= 0 when →0,
so he expansion
(2)
and he polynomials
E(α)
n
(
x
;
λ
)exis . Fo he case
λ
=
−
1,
clea ly 2
/
(
−e
+ 1)
≈ −
2
−1
when
→
0, so
E(α)
n
(
x
;
−
1) can be de ined only o
α= 0 (a i ial case) o a nega i e in ege .
Fo he au ho s, i is ex emely su p ising ha some kind o discussion as in
he p e ious pa ag aphs has no been included in he pape s o books ha de ine
and s udy he gene alized Apos ol-Be noulli, Apos ol-Eule and Apos ol-Genocchi
polynomials o o de
α∈C
( he abo e men ioned [
7
,
8
,
12
] and some o he ha
con inue wi h he s udy), es ic ing he de ini ion o he polynomials o he cases
when hey exis . Ins ead, many p ope ies o nonexis en polynomials, and o mal
ela ions be ween hem, can be ound in he ma hema ical li e a u e. The same
can be said o many u he gene aliza ions o
(1)
,
(2)
and
(3)
wi h he addi ion
o some ex a pa ame e s. Fo ins ance, he so-called gene alized Apos ol ype
polynomials F(α)
n(x;λ;µ;ν) ha we can ind in [12, p. 101] de ined ia
2µ ν
λe + 1αex =
∞
X
n=0
F(α)
n(x;λ;µ;ν) n
n!,
whose aim is o gi e a uni ied p esen a ion o
(1)
,
(2)
and
(3)
, and some o he s
ha appea in he same book, o he gene alized Apos ol-Be noulli polynomials o
le el m∈Nand o de α∈C ha , as we can see in [5], a e de ined ia
 m
λe −Pm
l=0 l/l!αex =
∞
X
n=0
B[m−1,α]
n(x;λ) n
n!
(o cou se, simila de ini ions a e gi en o Eule o Genocchi polynomials), ha
a e gene aliza ions o he p eexis ing cases co esponding o
λ
= 1 and
α
= 1. A
ecen e iew ha deals wi h many o hese gene aliza ions and uni ica ions is [
3
].
Fo una ely, many o he p ope ies and ela ions ha appea in all hese pape s
o books a e alid when he polynomials do exis , al hough his equi es a sui able
es ic ion o he pa ame e s. One o he aims in his a icle is o ask o mo e igo
in u u e pape s.
2. Rela ionship and educ ion
Once we ha e in oduced he pa ame e
λ
, he h ee amilies o polynomials
(1)
,
(2)
and
(3)
a e closely ela ed. Le us see his, as well as wha o do o a oid
essen ially edundan de ini ions.
As we ha e explained,
(1)
and
(3)
can be used only o
α∈N∪ {
0
}
(excep in
he case
λ
= 1 in
(1)
o
λ
=
−
1in
(3)
), and ha
(2)
is alid o gene al
α∈C
(excep in he case
λ
=
−
1). Now, we a e going o see ha he polynomials ha

EXISTENCE AND REDUCTION OF POLYNOMIALS 161
a ise in
(1)
and
(3)
can be educed o he gene alized Apos ol-Eule polynomials
o o de αin (1).
Le
α∈N∪ {
0
}
and he gene alized Apos ol-Be noulli polynomials o o de
α
be de ined as in (1), wi h λ6= 1. We can w i e
∞
X
n=0
B(α)
n(x;λ) n
n!=
λe −1αex = (−1)α2−α α2
(−λ)e + 1αex
= (−1)α2−α
∞
X
k=0
E(α)
k(x;−λ) k+α
k!(wi h he change n=k+α)
= (−1)α2−α
∞
X
n=α
E(α)
n−α(x;−λ) n
(n−α)!
= (−1)α2−α
∞
X
n=α
E(α)
n−α(x;−λ)n!
(n−α)!
n
n!.
Then, by using he uniqueness o Taylo expansions, we ha e he ollowing ela ion-
ships:
Theo em 1. Fo α∈N∪ {0}and λ6= 1, we ha e
B(α)
0(x;λ) = B(α)
1(x;λ) = · · · =B(α)
α−1(x;λ)=0
and
(4) B(α)
n(x;λ) = (−1)αn!
2α(n−α)!E(α)
n−α(x;−λ), n =α, α + 1, α + 2, . . .
In a simila way, le
α∈N∪ {
0
}
and he gene alized Apos ol-Genocchi polyno-
mials o o de αbe de ined as in (3), wi h λ6=−1. We s a e he ollowing:
Theo em 2. Fo α∈N∪ {0}and λ6=−1, we ha e
G(α)
0(x;λ) = G(α)
1(x;λ) = · · · =G(α)
α−1(x;λ)=0
and
(5) G(α)
n(x;λ) = n!
(n−α)!E(α)
n−α(x;λ), n =α, α + 1, α + 2, . . .
On he o he hand, le us ecall ha he polynomials
E(α)
n
(
x
;
−
1) can be de ined
only o α= 0 o a nega i e in ege . In his case, we can w i e
∞
X
n=0
E(α)
n(x;−1) n
n!=2
−e + 1αex = (−2)α −α
e −1αex
= (−2)α
∞
X
k=0
B(α)
k(x; 1) k−α
k!(wi h he change n=k−α)
= (−2)α
∞
X
n=−α
B(α)
n+α(x; 1) n
(n+α)!
162 L.M. NAVAS, F.J. RUIZ AND J.L. VARONA
= (−2)α
∞
X
n=−α
B(α)
n+α(x; 1) n!
(n+α)!
n
n!,
so we ha e he ollowing:
Theo em 3. Fo α= 0 o a nega i e in ege , we ha e
E(α)
0(x;−1) = E(α)
1(x;−1) = · · · =E(α)
−α−1(x;−1) = 0
and
(6) E(α)
n(x;−1) = (−2)αn!
(n+α)!B(α)
n+α(x; 1) , n =−α, −α+ 1,−α+ 2, . . .
We ha e seen in
(4)
and
(5)
ha he polynomials
B(α)
n
(
x
;
λ
)(wi h
λ6
= 1) and
G(α)
n
(
x
;
λ
)(wi h
λ6
=
−
1) can be exp essed, up o a mul iplica i e cons an , in e ms
o he gene alized Apos ol-Eule polynomials o o de
α
. Mo eo e , he pa ame e
α
in
(2)
can be any
α∈C
(excep when
λ
=
−
1), wi hou he es ic ion
α∈N∪ {
0
}
in
(1)
and
(3)
. Then, we can ake he gene alized Apos ol-Eule polynomials o
o de α∈Cde ined as
(7) 2
λe + 1αex =
∞
X
n=0
E(α)
n(x;λ) n
n!, λ ∈C {−1},
as he “main amily”, and conside
B(α)
n
(
x
;
λ
)and
G(α)
n
(
x
;
λ
)( o
λ6
= 1 in he i s
case and
λ6
=
−
1in he second) as supe luous a ia ions. To co e he “excep ional
cases”, and aking in o accoun ha
G(α)
n
(
x
;
−
1) = (
−
2)
αB(α)
n
(
x
; 1) and
(6)
, we
mus also de ine
(8) 
e −1αex =
∞
X
n=0
B(α)
n(x; 1) n
n!=
∞
X
n=0
B(α)
n(x) n
n!,
which a e he classical Be noulli polynomials (
α
= 1) o he Nø lund polynomials
(
α∈C {
1
}
). The polynomials de ined in
(7)
and
(8)
, bo h o
α∈C
, co e he
en i e ange o “ alid polynomials”.
We can highligh his classi ica ion by means o a sui able de ini ion:
De ini ion 4.
Le
λ, α ∈C
. We call he polynomials de ined by
(7)
when
λ6
=
−
1,
and by (8) when λ=−1, he Apos ol-like polynomials o o de (λ, α).
Finally, we a e going o p o e ha , wi h he de ini ions in
(7)
and
(8)
o ou
p ocess o educ ion, i no longe happens ha he
n
h polynomial can ha e
deg ee di e en om
n
. Ac ually, his is a ou ine a gumen once we es ablish ha
E(α)
n
(
x
;
λ
)and
B(α)
n
(
x
)a e non-null cons an s. Mo eo e , al hough we ha e e y
o en s a ed all along in his pape ha he coe icien s
E(α)
n
(
x
;
λ
)and
B(α)
n
(
x
)o
he analy ic expansions
(7)
and
(8)
a e polynomials on he a iable
x
, we ha e no
p o ed i in his pape .
To achie e hese goals, i is enough o gi e a simple esul ha is a well-known
p ocedu e in he heo y o Appell sequences, and ha we include o comple eness.
EXISTENCE AND REDUCTION OF POLYNOMIALS 163
Lemma 5.
Gi en an analy ic unc ion
A
(
)in an open disk a ound
= 0 such
ha A(0) 6= 0, le us conside he expansion
(9) A( )ex =
∞
X
n=0
n(x) n
n!.
Then 0(x) = A(0) and
(10) 0
n+1(x)=(n+ 1) n(x), n ∈N∪ {0}.
Mo eo e , n(x)is a polynomial o deg ee n o e e y n∈N∪ {0}.
P oo .
Fi s ly, by subs i u ing
= 0 in
(9)
we ge
A
(0)
·
1 =
0
(
x
)+0, so
0
(
x
) =
A
(0), a non-null cons an . Secondly, i we di e en ia e
(9)
wi h espec o
x, we ob ain
A( ) ex =
∞
X
n=1
0
n(x) n
n!,
so
∞
X
k=0
k(x) k
k!=A( )ex =
∞
X
n=1
0
n(x) n−1
n!
=
∞
X
k=0
0
k+1(x) k
(k+ 1)! =
∞
X
k=0
1
k+ 1 0
k+1(x) k
k!,
and
(10)
ollows by he uniqueness o he analy ic expansions. Finally, ha
n
(
x
)
is always a polynomial o deg ee
n
is a di ec consequence o
0
(
x
) =
cons an 6
= 0
and (10). 
Ac ually, ano he s anda d way o p o e ha
n
(
x
)is a polynomial o deg ee
n
is as ollows. As
A
(
)is analy ic a ound
= 0 and
A
(0)
6
= 0, also 1
/A
(
)is analy ic
a ound
= 0, so we will ha e 1
/A
(
) =
P∞
n=0 bn n/n
!, wi h
b06
= 0. Then, we can
w i e (9) as
X
n=0
xn n
n!=∞
X
n=0
bn
n
n! ∞
X
n=0
n(x) n
n!=
∞
X
n=0 n
X
k=0 n
kbn−k k(x) n
n!.
Consequen ly,
xn=
n
X
k=0 n
kbn−k k(x),
which, mo eo e , p o ides a ecu ence ela ion o ge he polynomials.
Le us apply Lemma 5 o bo h
(7)
and
(8)
. In
(7)
, he analy ic unc ion
A
(
)is
A( ) = 2
λe + 1α
so, in pa icula ,
E(α)
0(x;λ) = 2
λe0+ 1α=2
λ+ 1α6= 0 .
164 L.M. NAVAS, F.J. RUIZ AND J.L. VARONA
Simila ly, he analy ic unc ion in (8) is
A( ) = 
e −1α=1
1 + /2 + 2/3! + · · ·α
and hus
B(α)
0(x) = 1
1+0α= 1 .
In his way, we ha e p o ed he ollowing:
Theo em 6.
Fo
α∈C
, le us de ine
E(α)
n
(
x
;
λ
)( o
λ∈C {−
1
}
) and
B(α)
n
(
x
)
as in (7) and (8), espec i ely. Then,
E(α)
0(x;λ) = 2
λ+ 1α6= 0 ,B(α)
0(x)=1,
and
d
dxE(α)
n+1(x;λ)=(n+ 1)E(α)
n(x;λ),d
dxB(α)
n+1(x)=(n+ 1)B(α)
n(x),
o e e y
n∈N∪ {
0
}
. In pa icula ,
E(α)
n
(
x
;
λ
)and
B(α)
n
(
x
)a e polynomials o
deg ee non he a iable x.
Re e ences
[1]
Apos ol, T., On he Le ch Ze a unc ion, Addendum, Paci ic J. Ma h.
1
(1951), 161–167,
Paci ic J. Ma h. 2(1952), 10.
[2]
Bayad, A., Fou ie expansions o Apos ol-Be noulli, Apos ol-Eule and Apos ol-Genocchi
polynomials, Ma h. Comp. 80 (2011), 2219–2221.
[3]
He nández-Llanos, P., Quin ana, Y., U ieles, A., Abou ex ensions o gene alized Apos ol- ype
polynomials, Resul s Ma h. 68 (2015), 203–225.
[4]
Ho adam, A.F., Genocchi polynomials, Applica ions o Fibonacci Numbe s (G. E. Be gum, A.
N. Philippou, Ho adam, A. F., eds.), ol. 4, Kluwe , 1991, pp. 145–166.
[5]
Ku , B., Some ela ionships be ween he gene alized Apos ol-Be noulli and Apos ol-Eule
polynomials, Tu kish Jou nal o Analysis and Numbe Theo y 1(2013), 54–58.
[6]
Luo, Q.-M., Fou ie expansions and in eg al ep esen a ions o he Apos ol-Be noulli and
Apos ol-Eule polynomials, Ma h. Comp. 78 (2009), 2193–2208.
[7]
Luo, Q.-M., S i as a a, H.M., Some gene aliza ions o he Apos ol-Be noulli and
Apos ol-Eule polynomials, J. Ma h. Anal. Appl. 308 (2005), 290–302.
[8]
Luo, Q.-M., S i as a a, H.M., Some gene aliza ions o he Apos ol-Genocchi polynomials
and he S i ling numbe s o he second kind, Appl. Ma h. Compu . 217 (2011), 5702–5728.
[9]
Na as, L.M., Ruiz, F.J., Va ona, J.L., Asymp o ic es ima es o Apos ol-Be noulli and
Apos ol-Eule polynomials, Ma h. Comp. 81 (2012), 1707–1722.
[10] Nø lund, N.E., Mémoi e su les polynômes de Be noulli, Ac a Ma h. 43 (1922), 121–196.
[11]
Nø lund, N.E., Vo lesungen übe Di e enzen echnung, 1s ed., Sp inge -Ve lag,
Be lin-Heidelbe g, 1924.
[12]
S i as a a, H.M., Choi, J., Ze a and
q
-ze a unc ions and associa ed se ies and in eg als,
Else ie , 2012.
[13]
S i as a a, H.M., Ku , B., Simsek, Y., Co igendum: Some amilies o Genocchi ype
polynomials and hei in e pola ion unc ions, In eg al T ans o ms Spec. Func .
23
(2012),
939–940.