1 Fibonacci polynomials and central factorial numbers Johann Cigler [email protected]Abstract I consider some bases of the vector space of polynomials which are defined by Fibonacci and Lucas polynomials and compute the matrices of corresponding basis transformations. It turns out that these matrices are intimately related to Stirling numbers and central factorial numbers and also to Bernoulli numbers, Genocchi numbers and tangent numbers. There is also a close connection with the Akiyama-Tanigawa algorithm. Since such numbers have been extensively studied it would be no surprise if some of these results are already known, but hidden in the literature. I would therefore be very grateful for hints to relevant papers or books or for other comments. 1. Introduction We consider (a variant of) the Fibonacci polynomials defined by 1 2 0 1 () . n k n k n k F s s k − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = − − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑ (1.1) They satisfy the recursion 1 2 () () () n n n F s F s sF s − − = + with initial values 0 () 0 F s = and 1 () 1 Fs = . The first terms are ሺ0,1,1,1 ,ݏ12 ,ݏ13 ݏ ݏଶ ,14 ݏ3 ݏଶ ,15 ݏ6 ݏଶ ݏଷ ,16 ݏ 10 ݏଶ 4 ݏଷ ,…ሻ. The corresponding Lucas polynomials are defined by 2 0 () n k n k n k n L s s k n k ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = − ⎛ ⎞ = ⎜ ⎟ − ⎝ ⎠ ∑ (1.2) and satisfy the same recurrence as the Fibonacci polynomials, but with initial values 0 () 2 L s = and 1 () 1. L s = The first terms of this sequence are ሺ2,1,1 2 ,ݏ13 ,ݏ14 ݏ2 ݏଶ ,15 ݏ5 ݏଶ ,16 ݏ9 ݏଶ 2 ݏଷ , … ሻ. Let
47
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1
Fibonacci polynomials and central factorial numbers
I consider some bases of the vector space of polynomials which are defined by Fibonacci and Lucas polynomials and compute the matrices of corresponding basis transformations. It turns out that these matrices are intimately related to Stirling numbers and central factorial numbers and also to Bernoulli numbers, Genocchi numbers and tangent numbers. There is also a close connection with the Akiyama-Tanigawa algorithm. Since such numbers have been extensively studied it would be no surprise if some of these results are already known, but hidden in the literature. I would therefore be very grateful for hints to relevant papers or books or for other comments.
1. Introduction
We consider (a variant of) the Fibonacci polynomials defined by
12
0
1( ) .
n
kn
k
n kF s s
k
−⎢ ⎥⎢ ⎥⎣ ⎦
=
− −⎛ ⎞= ⎜ ⎟
⎝ ⎠∑ (1.1)
They satisfy the recursion 1 2( ) ( ) ( )n n nF s F s sF s− −= + with initial values 0 ( ) 0F s = and 1( ) 1F s = .
The corresponding Lucas polynomials are defined by
2
0
( )
n
kn
k
n knL s skn k
⎢ ⎥⎢ ⎥⎣ ⎦
=
−⎛ ⎞= ⎜ ⎟− ⎝ ⎠∑ (1.2)
and satisfy the same recurrence as the Fibonacci polynomials, but with initial values 0 ( ) 2L s = and 1( ) 1.L s = The first terms of this sequence are
2,1,1 2 , 1 3 , 1 4 2 , 1 5 5 , 1 6 9 2 ,… .
Let
2
1 1 42
sα + += (1.3)
and
1 1 42
sβ − += (1.4)
be the roots of the equation 2 0.z z s− − =
Then for 14
s ≠ − the well-known Binet formulae give
( )n n
nF s α βα β−
=−
(1.5)
and
( ) .n nnL s α β= + (1.6)
For 14
s = − it is easily verified that
1
1 .4 2n n
nF −⎛ ⎞− =⎜ ⎟⎝ ⎠
(1.7)
It is also well known that 1 1( ) ( ) ( ).n n nL s F s sF s+ −= + This follows immediately from s αβ= − and
( ) ( ) ( )1 1 1 1 .n n n n n nα β αβ α β α α β β α β+ + − −− − − = − + −
Since 2 1 2 2 2 2 1deg ( ) deg ( ) deg ( ) deg ( )n n n nF s F s L s L s n+ + += = = = each of the sets
{ } { } { } { }2 1 2 2 2 2 1( ) , ( ) , ( ) , ( )n n n nF s F s L s L s+ + + is a basis for the vector space of polynomials in .s
I am interested in the matrices which transform one basis into another. Their entries are related to Bernoulli numbers, Genocchi numbers and tangent numbers. Furthermore there are interesting connections with Stirling numbers and central factorial numbers.
The Genocchi numbers ( ) ( )2 11,1,3,17,155,2073,38227,929569,n n
G≥= (cf. OEIS A110501 )
and their relatives ( )1( ) 1, 1,0,1,0, 3,0,17,0, 155,n ng ≥ = − − − (cf. OEIS A036968) are defined by
their exponential generating function
2
21 1
2 1 ( 1) ,1 1 ! (2 )!
z n nn
n nz zn n
z e z zz z g z Ge e n n≥ ≥
−= + = = + −
+ + ∑ ∑ (1.8)
3
and the tangent numbers ( ) ( )2 1 1,2,16, 272,7936,353792, 22368256,nT + = (cf. OEIS A000182)
are defined by
2 2 1
2 120
1 ( 1) .1 (2 1)!
z kk
kzk
e zTe k
+
+≥
−= −
+ +∑ (1.9)
Note that by comparing coefficients we get
1 2 2 2 2 2 12 1( 1) .
2 2 2 2 2k k k k
k
g G Tk k
− + + ++− = =
+ + (1.10)
It is also well-known that 22 2( 1) 2(1 2 )n n
n nG B= − − , where ( )nB is the sequence of Bernoulli
numbers defined by
0
n
n kk
nB B
k=
⎛ ⎞= ⎜ ⎟
⎝ ⎠∑ (1.11)
for 1n > with 0 1.B = The list of Bernoulli numbers begins with
1, , , 0, , 0, , 0, , 0, , 0, , 0, , 0, ,…
For later uses let us also recall the generating function of the Bernoulli numbers
0
.! 1
n
n zn
z zBn e≥
=−∑ (1.12)
2. Connection constants
The following theorem gives an explicit computation of some basis transformations.
Theorem 2.1
The bases ( )2 2nF + and ( )2 1nF + are connected by
2 2 22 2 2 1
0
2 2( ) ( 1) ,
22 1
nn k n k
n kk
nGF s Fkk
− − ++ +
=
+⎛ ⎞= − ⎜ ⎟+ ⎝ ⎠∑ (2.1)
and
2 22 1 2 2
0
2 1( ) ( ).
2 1 1
nn k
n kk
n BF s F sk k
−+ +
=
+⎛ ⎞= ⎜ ⎟+ +⎝ ⎠∑ (2.2)
4
The bases consisting of Lucas polynomials are connected by
2 2 1 2 2 22 1 2 22 2 1
0 0
2 1 2 1( ) ( 1) ( ) ( 1) ( )
2 22 2 2 2
n nn k n kn k n k
n k kn kk k
n nT GL s L s L sk kn k
− −− + − ++ − +
= =
+ +⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟− +⎝ ⎠ ⎝ ⎠∑ ∑ (2.3)
and
2 22 2 1
0
2( ) 2 ( ).
2 2 1
nn j
n jj
n BL s L s
j j−
+=
⎛ ⎞= ⎜ ⎟ +⎝ ⎠∑ (2.4)
Proof
1) Since 1α β+ = we have
( )(1 ) (1 ) .z z z z z z ze e e e e e eα β β α α β− − − −− = − = − − (2.5)
This gives
0 0
( ) ( )( )! !
n z nn n
n n
F s F sz e zn n≥ ≥
= − −∑ ∑ (2.6)
or
2 1
2 11 ( ) ( )
2 ! (2 1)!
z n n
n nn n
e z zF s F sn n
− +
++
=+∑ ∑ (2.7)
This reduces our task to finding the matrix which corresponds to the operator "multiplication by 1
2
ze−+ " of the corresponding exponential generating functions.
Comparing coefficients we see that for 0( )n nx x ≥= and 0( )n ny y ≥=
1
2 ! !
z n n
n nn n
e z zx yn n
−+=∑ ∑ (2.8)
is equivalent with y Mx=
where ( )( , )M m i j= with 1( , ) ( 1)2
n k nm n k
k− ⎛ ⎞
= − ⎜ ⎟⎝ ⎠
for ,k n< ( , ) 1m n n = and ( , ) 0m n k = for
.k n>
For example
5
6M =
1 0 0 0 0 01 0 0 0 0
1 1 0 0 0
1 0 0
2 3 2 1 0
5 5 1
.
The inverse of (2.8) is
11
0
2 ( ) ( 1)! 1 ! ( 1) ! ! ! 1
n n j k n nn k n k
n n j k kzn n j k n k
nz z z z z gx y g y ykn e n j j k n n k
− − ++−
=
⎛ ⎞−= = = − ⎜ ⎟+ + − + ⎝ ⎠
∑ ∑ ∑ ∑ ∑ ∑ , thus
1
0
( 1) .1
nn k n k
n kk
ngx ykn k
− − +
=
⎛ ⎞= − ⎜ ⎟− + ⎝ ⎠∑ (2.9)
For example
16M − =
1 0 0 0 0 01 0 0 0 0
0 1 1 0 0 00 1 0 0
0 1 0 2 1 00 0 1
.
Thus (2.7) implies
1 2 22 1( ) ( 1)
2 12 2n n k
n k
ngF s Fkn k
− + −−
⎛ ⎞= − ⎜ ⎟−+ − ⎝ ⎠∑
and we get as special case
2 2 2 2 2 22 2 2 1 2 1
0 0
2 2 2 2( ) ( 1) ,
2 1 22 2 2 2 1
nn kn k n k
n k kk k
n ng GF s F Fk kn k k
−− + − ++ + +
≥ =
+ +⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟+− + +⎝ ⎠ ⎝ ⎠∑ ∑
i.e. (2.1).
From
( )(1 ) (1 )z z z z z z ze e e e e e eα β β α α β− − − −+ = + = + (2.10)
we get in the same way as before
6
2
21 ( ) ( ) .
2 ! (2 )!
z n n
n nn n
e z zL s L sn n
−+=∑ ∑ (2.11)
In this case (2.9) implies (2.3).
2) Another consequence of (2.5) is
( ) 22
0 0
( ) ( )1 2 .! (2 )!
z n nn n
n n
F s F se z zn n
−
≥ ≥
− =∑ ∑ (2.12)
Now
11
0 1
(1 ) ( ) ( 1) ( ) ( 1)! ! !
k n nnz n j
k n j n
nz z ze x k x j y njk n n
−− − −
= ≥
⎛ ⎞− = − = −⎜ ⎟
⎝ ⎠∑ ∑ ∑ ∑ is equivalent with
0
1( ) ( 1) ( ).
nn j
j
ny n x j
j−
=
+⎛ ⎞= − ⎜ ⎟
⎝ ⎠∑ (2.13)
The inverse is
11( ) ( 1) ( 1)! (1 ) ! ! !
k n k jk
zk n n n
z z b z zx k y n y jk e n k j
−
−= − = −−∑ ∑ ∑ ∑
where ( ) nb n B= for 1n ≠ and 1(1) .2
b = This follows from
.! ! 1 1 1
n n z
n n z z zn n
z z z ze zb B z zn n e e e−= + = + = =
− − −∑ ∑
This implies
0
( ) ( ).1
nn j
j
n bx n y j
j j−
=
⎛ ⎞= ⎜ ⎟ +⎝ ⎠∑ (2.14)
From (2.12) we see that with ( ) ( )nx n F s= and 2(2 1) 2 ( ), (2 ) 0,ny n F s y n− = = we get
If we choose 2( )w n n= we get the central factorial numbers ( , ) ( , )wt n k s n k= of the first kind and the central factorial numbers ( , ) ( , )wT n k S n k= of the second kind respectively.
13
These numbers have been introduced in [12] with a different notation. Further results can be found in [14], Exercise 5.8.
The following tables show the upper part of the matrices of central factorial numbers. See also OEIS A036969.
There are some interesting relations between central factorial numbers and Legendre-Stirling numbers which are analogous to the corresponding results about Stirling numbers.
Theorem 3.6
0
2( , ) ( 1, 1)
n
j
n jLS j k T n k
j=
−⎛ ⎞= + +⎜ ⎟
⎝ ⎠∑ (3.17)
and
0
2 1( , ) ( 1) ( 1, 1)
n
j
n jLS j k k T n k
j=
+ −⎛ ⎞= + + +⎜ ⎟
⎝ ⎠∑ (3.18)
15
Proof
For 1n k< + all terms vanish. For 1n k= + we have
0
2 1 2 1( , ) ( 1) ( 1) ( 1, 1)
k
j
k j k kLS j k k k T k k
j k=
+ − + −⎛ ⎞ ⎛ ⎞= = + = + + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠∑
and
0
2( , ) 1 ( 1, 1).
k
j
k j kLS j k T k k
j k=
−⎛ ⎞ ⎛ ⎞= = = + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠∑
Now assume that (3.17) and (3.18) are already known up to 1.n − Then
If we define a linear functional λ on the vector space of polynomials in s by
( )2 1( ) [ 0],nF s nλ + = = (4.3)
we get
2
21( ( )) .
(2 )! 1
n z
n z
z eF s zn e
λ⎡ ⎤ −
= ⎢ ⎥ +⎣ ⎦ (4.4)
Using (1.8) this gives (cf. [5],[3])
12 2( ( )) ( 1) .n
n nF s Gλ −= − (4.5)
By comparing coefficients we get again (cf. [3])
2 2 2 10
( ) ( , ) ( )n
n kk
F s a n k F s+ +=
= ∑ (4.6)
with
22
2 2 2
2 21( , ) ( 1) .22 1
n kn k
na n k G
kk−
− +
+⎛ ⎞= − ⎜ ⎟+ ⎝ ⎠
(4.7)
We call the infinite matrix ( ) , 0( , )
n kA a n k
≥= and the finite parts ( ) 1
, 0( , ) n
n i jA a i j −
== Genocchi
matrices.
E.g.
5
1 0 0 0 01 2 0 0 0
.3 5 3 0 017 28 14 4 0
155 255 126 30 5
A
⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟= −⎜ ⎟− −⎜ ⎟⎜ ⎟− −⎝ ⎠
(4.8)
It is clear that A is the matrix version of "multiplication with 11
z
z
ee−+
" of a certain kind of
generating functions. More precisely we have
Proposition 4.1
Let
0
1
2
xx
x x
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
,
0
2
2
yy
y y
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
and ( ) , 0( , ) .
n kA a n k
≥=
Then y Ax= is equivalent with
2 2
2 1
0 0
1 .(2 2)! 1 (2 1)!
n znn
n zn n
z e xy zn e n
++
≥ ≥
−=
+ + +∑ ∑ (4.9)
By considering the highest powers of s we see from (1.1) that ( , ) 1.a k k k= + Therefore the eigenvalues of A are 1,2,3, .
For later uses we note that
0
( , ) 1.n
k
a n k=
=∑ (4.10)
This follows from (4.6) for 0.s =
23
For special values of s (4.6) gives some interesting identities. Consider for example 1.s = − Here ( )0( ( )) 0,1,1,0, 1, 1,n nF s ≥ = − − is periodic with period 6.
If we delete the first row and first column we get the matrix
( ) ( ) ( )( )( ) 1
, 0 , 0 , 0( 1, 1) [ ]( 2) ( 1, 1) .
i j i j i jLS i j i j j LS i j
−
≥ ≥ ≥+ + = + + + (4.21)
27
The left-upper part begins with
2 0 0 0 02 3 0 0 08 8 4 0 056 56 20 5 0
608 608 216 40 6
.
Theorem 4.2 implies that the eigenvectors of ( ) , 0( , )
i jA a i j
≥= with respect to the eigenvalue k
are
(1, )(2, )
( ) .(3, )(4, )
T kT k
x k T kT k
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
(4.22)
This means that 0
( , ) ( 1, ) ( 1, ).j
a n j T j k kT n k≥
+ = +∑
This is equivalent with
2 1 2
1 1
1 ( , ) ( , ) .1 (2 1)! (2 )!
zn n
zn n
e T n k T n kz k ze n n
−
≥ ≥
−=
+ −∑ ∑ (4.23)
Let
2
0
( , )( ) .(2 )!
nk
n
T n kT z zn≥
= ∑ (4.24)
Then (4.23) can be written as
1 ( ) ( ).1
z
k kz
e T z kT ze− ′ =+
(4.25)
This result can also be obtained from Proposition 3.4.
28
Corollary 4.4
Define linear functionals kϕ by ( )2 1( ) ( , )k nF s T n kϕ − = for 1.k ≥ Then ( )2 ( ) ( , ).k nF s kT n kϕ =
Proof
This follows from (4.2).
An explicit expression for these functionals gives
Theorem 4.5
The following formulae hold:
1( ) ( , ),nk s LS n kϕ + = (4.26)
( )1
1 2 10
2 2( ) ( , ) ( , 1)
n
k nj
n jF s LS j k T n k
jϕ
−
+ −=
− −⎛ ⎞= = +⎜ ⎟
⎝ ⎠∑ (4.27)
( )1
1 20
2 1( ) ( , ) ( 1) ( , 1).
n
k nj
n jF s LS j k k T n k
jϕ
−
+=
− −⎛ ⎞= = + +⎜ ⎟
⎝ ⎠∑ (4.28)
Proof
This is an immediate consequence of Theorem 3.6.
For our next results we need the square of .A
For example
25A =
1 0 0 0 03 4 0 0 0
17 25 9 0 0155 238 98 16 0
2073 3255 1428 270 25
.
Theorem 4.6
The square of nA is given by
( ) 12, 0
( , )[ ] nn i j
A aa i j j i −
== ≤ (4.29)
with
29
2 2 4
2 2 2( , ) ( 1) .2 (2 1)( 2 )
n kn k
n n kaa n k Gk k n k
−− +
+⎛ ⎞ + += − ⎜ ⎟ + + −⎝ ⎠
(4.30)
Proof
Let 2 1
0
( ) .(2 1)!
nn
n
xf z zn
+
≥
=+∑
Since y Ax= is equivalent with 2 2
0
1 ( )(2 2)! 1
n z
n zn
z ey f zn e
+
≥
−=
+ +∑ we see that 2z Ay A x= = is
equivalent with
2 2 1
1 11 1
2
1 1 1 ( )(2 )! 1 (2 1)! 1 1
1 1 1( ) ( )1 1 1
n z n z z
n nz z zn n
z z z
z z z
z e z e ez y f zn e n e e
e e ef z f ze e e
−
− −≥ ≥
′⎛ ⎞− − −= = ⎜ ⎟+ − + +⎝ ⎠
′⎛ ⎞ ⎛ ⎞− − −′= +⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
∑ ∑
(4.31)
Since
2 2
1 2 2
1
1 ( 1)1 1 (2 )!
z nn n
zn
e G ze n n
+ +
≥
⎛ ⎞−= −⎜ ⎟+ +⎝ ⎠∑ (4.32)
and
2 3
1 2
1
1 1 1 ( 1) ,1 1 1 2 (2 3)!
z z z nn n
z z zn
e e e G ze e e n n
−−
≥
′ ′′⎛ ⎞ ⎛ ⎞− − −− = = −⎜ ⎟ ⎜ ⎟+ + + −⎝ ⎠ ⎝ ⎠
∑ (4.33)
we get
2 2
2 2 4
1
21 ( ) ( 1) ( )2 21 (2 )! 2
z n nn k n k
zn k
ne z Gf z x kke n n k
− − +
=
⎛ ⎞ ⎛ ⎞− ′ = −⎜ ⎟ ⎜ ⎟−+ − +⎝ ⎠⎝ ⎠∑ ∑ (4.34)
and
2
2 2 4
0
21 ( ) ( 1) ( ).2 11 (2 )! 2 2 4
z n nn k n k
zn k
ne z Gf z x kke n n k
− − +
=
′′⎛ ⎞ ⎛ ⎞−− = −⎜ ⎟ ⎜ ⎟−+ − +⎝ ⎠⎝ ⎠
∑ ∑ (4.35)
From (4.34) and (4.35) and using (4.31) we get immediately (4.30).
It only remains to prove (4.32).
30
This follows from
2 2
2 20
21 2 2
0
1 1 (2 1)1 2 1 2 ( 1)1 1 (2 )!(2 1)(2 2)
( 1) .1 (2 )!
z z nn
nz zn
nn n
n
e e n zGe e n n n
G zn n
+≥
− +
≥
′⎛ ⎞ ⎛ ⎞− − += − = − −⎜ ⎟ ⎜ ⎟+ + + +⎝ ⎠ ⎝ ⎠
= −+
∑
∑
Theorem 4.7 (Further properties of 2nA )
( ,0) ( 1,0)aa n a n= − + (4.36)
and for 1 1k n≤ ≤ −
( , ) ( , 1) ( 1, )aa n k a n k a n k= − − + (4.37)
Proof.
This follows from (4.30).
Intimately connected with the linear functional λ is the linear functional *λ defined by
*( ( )) ( ( )).n nF s sF sλ λ= − (4.38)
Since 2 1( ( )) ( ( )) ( ( ))n n nsF s F s F sλ λ λ+ += − we get
2 2 2 2 1 2 2( ( )) ( ( )) ( ( )) ( ( ))n n n nsF s F s F s F sλ λ λ λ+ + += − = for 0n > and
2 1 2 3 2 2 2 2( ( )) ( ( ) ) ( ( )) ( ( )).n n n nsF s F s F s F sλ λ λ λ+ + + += − = −
The sequence ( )( )* ( )nF sλ begins with
0,1,1, 1, 3,3,17, 17, 155,155,2073, 2073, 38227.
Therefore
*2 2 1( ( ) ( )) [ 0]n nF s F s nλ ++ = = (4.39)
and
31
( )*2 1 2 2 2 2( ( ) ( )) ( 1) .n
n n n nF s F s G Gλ − ++ = − + (4.40)
Thus we are led to consider the basis consisting of the polynomials 2 2 2 1( ) ( ).k kF s F s− −+
Here we get
Theorem 4.8
Let
10
( , ) [ ] ( , ).k
j
a n k k n a n j=
= ≤ ∑ (4.41)
Then
( )2 1 1 2 2 10
( ) ( , ) ( ) ( ) .n
n k kk
F s a n k F s F s+ +=
= +∑ (4.42)
The first entries of ( )1( , )a n k are
1 0 0 0 0 01 1 0 0 0 03 2 1 0 0 017 11 3 1 0 0
155 100 26 4 1 02073 1337 346 50 5 1
.
Proof
We know that 1
2 2 10
( ) ( 1, ) ( ).n
n kk
F s a n k F s−
+=
= −∑ Therefore
1
2 2 1 2 1 2 10
( ) ( ) ( 1, ) ( ) ( ).n
n n k nk
F s F s a n k F s F s−
+ + +=
+ = − +∑ This implies that in (4.42) the matrix ( )1( , )a i j
is the inverse of I B+ , where I denotes the identity matrix and ( , ) ( 1, )b n k a n k= − for .k n< We have to show that 1( , )a n k is given by (4.41).
It suffices to show that 0
( ) [ ] ( , ) .j
I B j i a i I=
⎛ ⎞+ ≤ =⎜ ⎟
⎝ ⎠∑
32
This means that for each k
( )1
( 1, ) ( ,0) ( ,1) ( , ) ( ,0) ( , ) [ ].n
j k
a n j a j a j a j k a n a n k n k−
=
− + + + + + = =∑
Let .k n< By definition 1
0
( 1, ) ( 1, ) ( , ).n
j
aa n k a n j a j k−
=
− = −∑ Therefore
1 1
0
( 1, ) ( 1, ) ( , ) ( 1, ) ( , ).n k
j k j
aa n i a n j a j i a n j a j i− −
= =
− − − = −∑ ∑
Using (4.37) and (4.10) the left-hand side is equivalent with
We used the fact that ( ) ( ,0) ( 1,0) 0.w ww k s k s k+ + =
44
For ( ) 1w n n= + and 1( )1
a nn
=+
this reduces to the original Akiyama-Tanigawa algorithm for
( )( )b n as shown in [11].
Now we give a list of some formulas.
For ( ) 1w n n= + , 1( )1
a nn
=+
we get from (2.16) and (3.10)
0
!( 1, 1)( 1) ( ).1
nj
j
jS n j b nj=
+ + − =+∑ (6.6)
Another proof of (6.6) can be found in [11], but I suspect that this result must be much older.
Formula (6.3) gives
0
!( 1, 1) ( ) ( 1) .( 1)
nn
j
ns n j b jn=
+ + = −+∑ (6.7)
For 2( ) ( 1)w n n= + , ( ) 1a n n= + we know from (4.7) and (4.16) that 2 2( ) ( 1) .nnc n G += −
This gives
( )21 12
1
( 1) ( 1) ( , ) ( 1)!n
n kn
k
G T n k k k− −
=
− = − −∑ (6.8)
and
21
( 1) ( , ) !( 1)!.n
n kk
k
t n k G n n−
=
− = −∑ (6.9)
The Akiyama-Tanigawa algorithm applied to (6.8) gives another method for computing the Genocchi numbers. Choose 2( ) ( 1)w n n= + and ( ) 1a n n= + in Theorem 7.2.
Then the left upper part of the corresponding matrix ( )( , )M m i j= is given by
For example for 3n = we have 24 6 84 5 4 15 17 36 (3!) .G G G+ + = + + = =
(4.21) and Lemma 4.3 give
( )22 3
0
( 1) ( 1, 1) ( 1)!n
n kn
k
LS n k k H−+
=
− + + + =∑ (6.12)
From (4.45) and (4.44) we deduce for 2( ) ( 2)w n n= + and ( ) 2a n n= +
2 2 2 40
( 1) ( , )( 1)!( 2)!n
n k wn n
k
S n k k k G G−+ +
=
− + + = +∑ (6.13)
and
46
( )2 2 2 40
( 1) ( , ) ( 1)!( 2)!.n
n k wk k
k
s n k G G n n−+ +
=
− + = + +∑ (6.14)
For ( ) 1w n n= + and 1( )1
a nn
=+
we get from (4.49)
2
20
( !)(2 1) ( 1) ( 1, 1).1
nj
nj
jn B T n jj=
+ = − + ++∑ (6.15)
Here the left upper part of the Akiyama-Tanigawa matrix begins with
1
.
From (5.6) and (5.10) we deduce
( )22 1
0
( 1) 4 ( , )(2 1) (2 1)!! .n
n k n kn
k
U n k k k T− −+
=
− + − =∑ (6.16)
Finally Proposition 5.3 gives
( )22
0
1( 1) ( , ) (2 1)!! .(2 1)4
nk
nkk
U n k k Bk=
− − =+∑ (6.17)
47
References
[1] S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J. 5 (2001), 327-351
[2] G.E. Andrews and L.L. Littlejohn, A combinatorial interpretation of the Legendre-Stirling numbers, PAMS 137 (2009), 2581-2590
[3] J. Cigler, q-Fibonacci polynomials and q-Genocchi numbers, arXiv:0908.1219
[4] A. de Médicis and P. Leroux, Generalized Stirling numbers, convolution formulae, and p,q-analogues, Can. J. Math. 47(3), 1995, 477-499
[5] D. Dumont and J. Zeng, Further results on the Euler and Genocchi numbers, Aequat. Math. 47(1994), 31-42
[6] D. Dumont and J. Zeng, Polynomes d'Euler et fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998), 387-410
[7] A. v. Ettingshausen, Vorlesungen über die höhere Mathematik, 1. Band, Verlag Carl Gerold, Wien 1827
[8] Y. Gelineau and J. Zeng, Combinatorial interpretations of the Jacobi-Stirling numbers, arXiv: 0905.2899
[9] Y. Inaba, Hyper-sums of powers of integers and the Akiyama-Tanigawa matrix, J. Integer Sequences 8 (2005), Article 05.2.7
[10] M. Kaneko, A recurrence formula for the Bernoulli numbers, Proc. Japan Acad. Ser. A Math.Sci. 71 (1995), 192-193
[11] M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences 3 (2000), Article 00.2.9
[12] J. Riordan, Combinatorial Identities, John Wiley, 1968
[13] L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandter Reihen, Sitzungsber. Münch. Akad. Math. Phys. Cl. 1877, 157-187