Pacific Journal of
Mathematics
SOME DETERMINANTS INVOLVING BERNOULLI AND
EULER NUMBERS OF HIGHER ORDER
F RANK R. O LSON
Vol. 5, No. 2
October 1955
SOME DETERMINANTS INVOLVING BERNOULLI AND EULER NUMBERS
OF HIGHER ORDER
FRANK R.
OLSON
1. Introduction. In this paper we evaluate certain determinants whose elements are the Bernoulli, Euler, and related numbers of higher order. In the notation of Norlund [ l , Chapter 6] these numbers may be defined as follows: the
Bernoulli numbers of order n by
t
(1.1)
\
__^ t
\n)
the related " β " numbers by
(12)
l — Y T (~D
V
—
the Euler numbers of order n by
(1.3)
and the "C"
(1.4)
(sec tr
= Z
(~1)V
TTTT W
^ L
=0),
numbers by
I
—
Ve' + i
(By n we denote an arbitrary complex number. When n = 1, we omit the upper
index in writing the numbers; for example, B ' = B .)
We evaluate determinants such as
\B%j
(*',/ = °» 1> ' , π ι )
\
for the Bernoulli numbers, and similar determinants for the other numbers. The
Received July 29, 1953.
Pacific J. Math. 5 (1955), 259-268
259
260
FRANK R. OLSON
proofs of these results follow from the evaluation of a determinant of a more
general nature; see (3.4), below. Finally, a number of applications are given.
n
( n
n)
2. Preliminaries and notation. The numbers B^ \ D 2 y\ EΪfJ, and c[
may
be expressed as polynomials in n of degree v [ 1, Chapter 6]; in particular,
Ώ(Π)
β
o
~
Γ)(rι) _ f?(n)
oo
~ 0
u
Λ
~
r(n) _ -i
U
0
~
i
φ
Although little is known about these polynomials, it will suffice for our purposes
to give explicitly the values of the coefficients of nv in each of the four cases
Considering first the Bernoulli numbers, we use the recursion formula [ 1 ,
p. 146]
(2.D
Let
and compare coefficients of nv on both sides of (2.1). We find that
But Bγ = - 1/2 and therefore bv = - δ J ^ / 2 . Since β^"^ = 1, the preceding leads
us recursively to
In a similar fashion the formula [ l , p . 146]
(2.3)
coupled with C^n = 1, permits us to write
DETERMINANTS INVOLVING BERNOULLI AND EULER NUMBERS
CJ;n) = ( ~ I ) v n
(2.4)
v
+ cv_ιnv
ι
261
+ -.+c0.
As for the Euler numbers, we consider the symbolic formula [ 1 , p. 124]
(E(n) +l )
(2.5)
2 v
+(E{n) - l )
2 v
2E[n'ι)
=
2v
in which, after expansion, exponents on the left side are degraded to subscripts.
Hence we have
(2.6)
4->+
( 2
' ; ) ( 2 t ; - 1 ) EM
+
...
=£
(n-O.
Writing
and
we see first that
^"J ~£(2vl)
+ terms of lower d e
=vevnV~l
Hence comparing coefficients of nv~ι
p
in (2.6) we have
_
~
i
^
2v
Since E
n
= 1, we obtain recursively
2V
(-2)%!
U
+ β v
'
i n
Next, from [ l , p. 129]
(2.8)
we find that
(Z)U) +l ) 2 t > + 1 - ( Z ) ( π ) - l ) 2 t ; + ι = 2 ( 2 ί ;
8ree '
262
FRANK R. OLSON
6/
t>!
We shall employ the difference operator Δ j = Δ for which
1
and Δ ' - Δ . Δ " - .
Δf(x)=f(x+d)-f{x)
We recall that if
f (x) =z avxv
+ av_ιxv~
+
+ α0 ,
+αno
(βi
then
(2.10)
v
v
Δ f(x)=aυd vl
3. Main results. Let
(3.1)
f ( x ) — an
n
x n + an n-ιxn~
+
and consider the determinant
(3.2)
( h 7 = ^> 1,
\f.{χ.)\
, m)«
This may be written as the product of the two determinants
0 ...0
ii
(3.3)
a
m,0
••• 0
m, I ' * ' am,m
1
1
...1
XQ
Xι
•• Xπ
X
X
a
0
l
The first determinant in (3.3) reduces simply to the product of fhe elements on
the main diagonal, and the second is the familiar Vandermond determinant.
Hence
(3.4)
|Λ ( * / ) | = Π aKk Π
A;=0
If we let
τ> s
(r,
S
= 0, 1, . . , m ) .
DETERMINANTS INVOLVING BERNOULLI AND EULER NUMBERS
263
then it follows from (3.4) and (2.2) that
/
(3.5)
\
m
I
I*,- ' Ί = Π
/c = 0
1 \k
- o ) Π <*r-*s>
V
(ίf/,rfs=0, l,...,m).
^ ' r > s
Application of (3.4) to (2.4), (2.7), and (2.9) yields results of a similar
nature for the C9 Ό9 and E numbers. Consequently we have:
THE
ORE M 1. For i, j = 0, 1, . . . , m ,
r>s
;
ι ==ΓT
(-lr
1 1
TΊ (x -x )
1 1
'
Γ
^
s
''
r>s
If we take %y = a + c? then we obtain:
COROLLARY 1. For i, ]^~ 0, 1,
, τn$ a and d constants,
/c = 0
(ii)
(ϋi)
'
J 7 λ
—
I
'
^
(iv)
/c=0
k
d\
x
6 /
264
FRANK R. OLSON
If we let
defined as in (3.1), then we can readily show by the above method that
f^x)
m
/c=0
Hence (3.6) implies
with like results for the other numbers.
We remark that the determinants of Corollary 1 may also be evaluated by a
succession of column subtractions.
4. Applications. We consider first the determinant
(4.1)
where B.'(x)
]B(.a^d)(x)\
U 7 = 0 , 1, . . . , m ; α, d constants),
is the Bernoulli polynomial of order n defined by [ 1 , p . 145]
(For x = 0 , B ^ n ) ( 0 ) = i 5 ^ ) , the Bernoulli number of order n.) Also, by [ l ,
p. 143],
B[n\x)^ £
Γ\xv'sB(sn).
s=o \ s '
Consequently
(4.2)
If we define
5=0
DETERMINANTS INVOLVING BERNOULLI AND EULER NUMBERS
I
J = 1 and |
265
) = 0 for / > £,
then the right member of (4.2) may be written as the product of the two determinants;
I
y
1
x"
T h e first d e t e r m i n a n t h a s v a l u e 1 a n d h e n c e , by Corollary l ( i ) ,
\B{a+Jd)(x)\
(4.3)
=Π
( - - )
kl
-
The Bernoulli polynomials may also be expressed in terms of the D numbers
by [ 1 , p. 130]
S=0
If in (4.3) we let % = hn9 h £ 1/2, then
[ v / a ]
(4.5)
β ( n ) (/ί r a )=
T
/vw
i r
11A — 1
n^^DrΛ
Since D^J may be written as a polynomial in n of degree s, and DQ = 1, it
follows readily from (4.4) that, expressed as a polynomial in n,
(4.6)
( n)
B v {hn)
v
= Ih - - ) n + terms of lower degree .
Consequently, using the same procedure that gave (3.4), we can show for α, d
fixed constants, i$ j ~ 0,1,
, m, that
m
(4.7)
α+
d)
I
\\k
| B j ' ' U ( α + / £ / ) ) I=Π ( A " - l
dkkl
-
For h = 0, (4.7) reduces to the case of Corollary l ( i ) . If h - 1/2 and v is odd,
then it follows from (4.4) that
266
FRANK R. OLSON
B(vn)(n/2) = 0.
v
Therefore for m >_ 1, the value of the determinant in (4.6) is zero. However,
if v is even, then
and
(4.7)'
/f=0
\
24
where in evaluating the second determinant we have applied Corollary l ( i i i ) .
Finally, it is of interest to point out that [ 1 , p. 4 ]
;=0
together with (2.2), (2.4), (2.7), (2.9), and (2.10) yield the recursion formulas
(4.8)
£
;=0
(4.9)
£
/=o
(4.10)
£
and
(4.11)
£
;=O
5. Some additional results. The above methods may also be applied to the
evaluation of determinants involving the classic orthogonal polynomials. We
consider first the Laguerre polynomials defined by [2, p. 97]
DETERMINANTS INVOLVING BERNOULLI AND EULER NUMBERS
(5.1)
267
L
Setting Cί = a + jd and writing (5.1) as a polynomial in / we have
in
{a+ d)
=j
L n J {x)
n
— + terms of lower degree .
n\
Consequently, as in § 3, we obtain
m-l
(5.2)
(a+ d)
k
Y2m(m ι)
= Γ Ί d =d
\L i (x)\
I
X
(i,/ =0,1, . . . , m - l ) ,
"
A.
k =0
For the Jacobi polynomials defined by [2, p. 67]
n
P * $ ( Ύ} — V*
n
*v=0
(^ S)
we set Cί = a + jd and hold /3 fixed. Then, as a polynomial in /
j
n
+ terms of lower degree .
— —
2n
n\
Hence, we find
(5.4)
Similarly
/ ,
(5.5)
6+
e
x
ί (x - D e l Vimim-l)
|φ / )()
|
j
We consider next, as a polynomial in /,
- j
'
}
£ί
(n-v)\
— I
I I
I
v\ \
2 / \
2 I
•+• terms of lower deeree
g
268
FRANK R. OLSON
j
n
n\ I
\(d+e)x+d-eV
2
+ terms of lower d e g r e e ,
which yields
(5.6)
\(d
+ e)x
e)x + d — e ]%
2
J
( h 7 = ^> 1,
, m — 1).
Finally, for (X = β, the Jacobi polynomials reduce to the ultraspherical polynomials P^Hx). It follows from (5.6) that
(5.7)
REFERENCES
1. N. E. Norlund, Differenzenrechnung, Berlin, 1924.
2. Gabor Szego, Orthogonal polynomials, New York, 1939.
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Pacific Journal of Mathematics
Vol. 5, No. 2
October, 1955
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L. Carlitz, The number of solutions of certain types of equations in a finite
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Arthur Pentland Dempster and Seymour Schuster, Constructions for poles
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