Solution to Problem 80-1: A determinant and an identity

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Solution to Problem 80-1: A determinant and an identity

Citation for published version (APA):

Lossers, O. P. (1981). Solution to Problem 80-1: A determinant and an identity. SIAM Review, 23(1), 105-107. https://doi.org/10.1137/1023015

DOI:

10.1137/1023015

Document status and date: Published: 01/01/1981

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PROBLEMS AND SOLUTIONS 105

Editorial note. Both conjectures have been proved valid by the proposers and

J.

BusToz(Arizona StateUniversity) andS. LINNArNMAA(UniversityofHelsinki) in their jointpaper, Improved trailing digitsestimatesapplied tooptimalcomputer arith-metic, J.

Assoc. Comput.

Mach., 26 (1979), pp. 716-730 (in particular, see pp.

721-723).

A

MatrixEigenvalue Problem

Problem 79-2, by G.EFROYMSON,

A. STEGER

and

S. STEINBERG

(University ofNew

Mexico).

Let

Mn

denote the n n matrixwhose(j, k)entry

Mn

(],

k)isgiven by o)

(J-1)(k-1)/4-,

1 j, k n,

where to e

2i/.

Determine allthe eigenvaluesof

M.

The matrix

_Mn

arisesin some work on finite Fouriertransforms.

(Previoussolutiongivenin thisJournal, 22 (1980),pp.99-100).

Comment by B. PaRLETr (University of California, Berkeley, California). Another solutionappearsinL. AUSLaNDERand

R. TOLIMIERI,

IScomputing with the

finite

Fourier

_transform

pureorapplied mathematics

?,

Bull.

Amer.

Math.Soc.,1 (1979),

pp. 855-857.

A

DeterminantandanIdentity

Problem 80-1, by A. V.BOYD(Universityofthe Witwatersrand, Johannesburg,South Africa).

(a)

Prove

that

det

IA,.s[

(-1)+(2

e-2)Bz,.,/(2n)!

c,t

"ca-

,+

qnt

(1)"

=0 =0 q0

Part (b)results from theelementaryidentity

[(xh

cschxh

)/2h ][e

x(t/h)-e

x(t-h)] xeXt

ql qo 0 0 q qo 0 q0 q,,- q,,-2 q where r, s 1,2,.

.,

n,

={1/(2r-2s+3)!,

s<-r+l,

Ars

O, s>r+l,

and

_Bn

isthe Bernoullinumberdefined by

Bt

e -1 =o n! (b) Provethat if n isodd,

tn

..

n

h2m_lB2m{(

+

h)n+l_2m (th)n+,_2m}.

,=o n-2m+l 2m

Solution by O. G. RUEHR (Michigan Technological University).

From

the given generating function for the Bernoulli numbers, the well-known expansion, x cschx-

_n0

q,x

,

q,,

(2--2")B2,/(2n)!,

iseasilyobtained.Part (a) is nowanimmediate consequenceof Wronski’s determinantfor the reciprocalof apower series,i.e.,

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106 PROBLEMS AND SOLUTIONS

uponmultiplying theseriescorrespondingtothe bracketed quantitiesand employing the formula

oO In/2]

m=0 n=l n=0 m=0

The stated result

(b)

is correct for nonnegative integers n, provided that

(n-

1)/2

is

replaced by

In/2]

asthe upperlimitof summation.

Remark.Thissolver recently rediscovered the following generalization of

Wron-ski’s determinant which had been quoted byMuir

[1]

withoutproof. Let

q.t"

_Z

c.(rn)t".

=0 =0

Then,c,(m)=

((-1)"/q

+")

det

[Ars(m)[,

where

l[(r-s

+

1)m +(s-

1)]qr-s+l/r,

Ar

0,

s=<r+l,

s>r+l.

The proof

[2],

dependsuponelementarypropertiesoflower Hessenbergmatricesand usesthe

J.

C.

P.

Millerformula

[3].

REFERENCES

1. T.MUIR, ATreatiseonthe Theory_ofDeterminants,Dover, NewYork,1960, p. 722. 2. O. G.RUEHR,AGeneralization_ofWronski’sDeterminant,unpublished memorandum.

3. P. HENRICI,Appliedand ComputationalComplexAnalysis, Wiley-Interscience,NewYork,1974, p. 42.

Solutionby O. P. LOSSERS (TechnischeHogeschool Eindhoven, the

Netherlands).

a) Let

D,

detlAr, [,

Do

1.

Then by expanding the determinant by the last column and iterating this procedure,we find (__1)i-1

D. _io

(2i

ii

!D.-i,

(1)

(--1)i-1

i=o(2i

+-!

Dn-i

=O"

Since one easily checks the assertion for n 1, 2 it is sufficient to see whether the asserted value of

_Dn

also satisfies

(1),

i.e., whether forn

>

0wehave

i

(2n+l)(22(’-/)

2)B2(,,-i 0.

(2)

i=o 2i+1

Forthewell-known Bernoullipolynomials,wehave

2"+1

(2n

+

_1’]

Bzn+l(t)

B2n+l-kt

k k=O k /

(3)

(

1 2n

+

1

)

t2

+

(2n+)

B2(n-i)

tzi+

2n =o 2i

+

(4)

PROBLEMS AND SOLUTIONS ₁₀₇

Nowsubstitute

t=1/2

andt= 1 in

(3)

anduse

B2n+l(1/2)

B2n+l(1) 0(n >0).

Then(2)follows by subtracting thetwoequations.

b)The right-hand side of(b)has.the form

aht

"-.

j-O

Clearly

_a.

0if]isodd. For ] even,wefind 1 n

+

1

(1- 2

’-)

B,,

a

_2]+

₁

=o 2m

whichis 0 if]

>

0 according to

(2).

Substituting h 0, wefind forthe right-handside

1_2-1

(t+h)"+_(t_h)

"+

lim

t".

h+O n+l h

Thisproves(b).

Also solved by C. GIVENS (Michigan Technological University), A. A. JAGERS (TechnischeHogeschool

Twente,

Enschede,the_{Netherlands), S. L. LEE (University}_of

Alberta) and theproposer.

Additionally,LEEprovides the generalization

det

_IArsl

(-1)k2"+k+1’/2det

IB,il,

where

Ars

1/[2(r-

s

+

k)

+ 1]!,

r,s 1,2,

,

n k

+

1, k 0, 1, (1/p! 0 forp

<

0),and

Bii

2(n+2-i-j), i,] 1,2,"

,

k, where B2, (2 2)B2,,/(2m)!.

His,

proofuses Sylvester’s identity andinduction.

n

AMatrixStability Problem

Problem

80-3*,

by K.

SOURISSEAU

(Universityof Minnesota) and M.

F.

DOHERTY (UniversityofMassachusetts).

Let

A

B1

C2

A2

B2

Solution to Problem 80-1: A determinant and an identity