They are orthogonal with respect to the functional L which is given by

π∞ x2

L(p(x)) = p(x) dx . (2.36)

2 ’∞ sinh2 (πx)

Explicitly, the orthogonality relation is

n! (n + 1)!4 (n + 2)!

L(pm (x)pn (x)) = δm,n . (2.37)

(2n + 2)! (2n + 3)!

In particular, L(1) = 1/6.

Now, by combining Theorems 11, 13, and 14, and by using an integral representation

of Bernoulli numbers (see [122, p. 75]),

√

∞ ’1

1 π 2

√ ν

Bν = z dz

√

2π ’1 sin πz

’∞ ’1

(if ν = 0 or ν = 1 then the path of integration is indented so that it avoids the

singularity z = 0, passing it on the negative side) we obtain without di¬culty the

desired determinant evaluation,

n n’1 n’i

i(i + 1)2 (i + 2)

1

n

(Bi+j+2 ) = (’1)( )

det 2

6 4(2i + 1)(2i + 3)

0¤i,j,¤n’1

i=1

n’1

i! (i + 1)!4 (i + 2)!

1

n

= (’1)( ) . (2.38)

2

6 (2i + 2)! (2i + 3)!

i=1

The general determinant evaluation which results from using continuous Hahn polyno-

mials with generic nonnegative integers a, b, c, d is worked out in [51, Sec. 5].

Let me mention that, given a Hankel determinant evaluation such as (2.38), one has

automatically proved a more general one, by means of the following simple fact (see for

example [121, p. 419]):

Lemma 15. Let x be an indeterminate. For any nonnegative integer n there holds

i+j

i+j

Ak xi+j’k

det (Ai+j ) = det . (2.39)

k

0¤i,j¤n’1 0¤i,j¤n’1

k=0

ADVANCED DETERMINANT CALCULUS 23

The idea of using continued fractions and/or orthogonal polynomials for the evalua-

tion of Hankel determinants has been also exploited in [1, 35, 113, 114, 115, 116]. Some

of these results are exhibited in Theorem 52. See the remarks after Theorem 52 for

pointers to further Hankel determinant evaluations.

2.8. Miscellaneous. This section is a collection of various further results on deter-

minant evaluation of the general sort, which I personally like, regardless whether they

may be more or less useful.

Let me begin with a result by Strehl and Wilf [173, Sec. II], a special case of which was

already in the seventies advertised by van der Poorten [131, Sec. 4] as ˜a determinant

evaluation that should be better known™. (For a generalization see [78].)

Lemma 16. Let f(x) be a formal power series. Then for any positive integer n there

holds

n

f (x) ( 2 )

i’1

d

(aj ’ ai ).

f(x)aj = f(x)a1 +···+an

det (2.40)

dx f(x)

1¤i,j¤n

1¤i<j¤n

By specializing, this result allows for the quick proof of various, sometimes surprising,

determinant evaluations, see Theorems 53 and 54.

An extremely beautiful determinant evaluation is the evaluation of the determinant

of the circulant matrix.

Theorem 17. Let n by a ¬xed positive integer, and let a0 , a1, . . . , an’1 be indetermi-

nates. Then

«

a0 a1 a2 . . . an’1

¬an’1 a0 a1 . . . an’2 · n’1

¬ ·

det ¬an’2 an’1 a0 . . . an’3 · = (a0 + ωi a1 + ω 2i a2 + · · · + ω(n’1)i an’1 ),

¬ ·

. . . . . . . . . . . . . . . . . . . . . . . . . i=0

a1 a2 a3 . . . a0

(2.41)

where ω is a primitive n-th root of unity.

Actually, the circulant determinant is just a very special case in a whole family of

determinants, called group determinants. This would bring us into the vast territory of

group representation theory, and is therefore beyond the scope of this article. It must

su¬ce to mention that the group determinants were in fact the cause of birth of group

representation theory (see [99] for a beautiful introduction into these matters).

The next theorem does actually not give the evaluation of a determinant, but of a

Pfa¬an. The Pfa¬an Pf(A) of a skew-symmetric (2n) — (2n) matrix A is de¬ned by

(’1)c(π)

Pf(A) = Aij ,

π (ij)∈π

where the sum is over all perfect matchings π of the complete graph on 2n vertices,

where c(π) is the crossing number of π, and where the product is over all edges (ij),

i < j, in the matching π (see e.g. [169, Sec. 2]). What links Pfa¬ans so closely to

24 C. KRATTENTHALER

determinants is (aside from similarity of de¬nitions) the fact that the Pfa¬an of a

skew-symmetric matrix is, up to sign, the square root of its determinant. That is,

det(A) = Pf(A)2 for any skew-symmetric (2n) — (2n) matrix A (cf. [169, Prop. 2.2]).9

Pfa¬ans play an important role, for example, in the enumeration of plane partitions,

due to the results by Laksov, Thorup and Lascoux [98, Appendix, Lemma (A.11)] and

Okada [123, Theorems 3 and 4] on sums of minors of a given matrix (a combinatorial

view as enumerating nonintersecting lattice paths with varying starting and/or ending

points has been given by Stembridge [169, Theorems 3.1, 3.2, and 4.1]), and their

generalization in form of the powerful minor summation formulas due to Ishikawa and

Wakayama [69, Theorems 2 and 3].

Exactly in this context, the context of enumeration of plane partitions, Gordon [58,

implicitly in Sec. 4, 5] (see also [169, proof of Theorem 7.1]) proved two extremely useful

reductions of Pfa¬ans to determinants.

Lemma 18. Let (gi ) be a sequence with the property g’i = gi , and let N be a positive

integer. Then

Pf 1¤i<j¤2N g± = det (gi’j + gi+j’1 ), (2.42)

1¤i,j¤N

’(j’i)<±¤j’i

and

g± j ¤ 2N + 1

’(j’i)<±¤j’i

= X · det (gi’j ’ gi+j ). (2.43)

Pf 1¤i<j¤2N +2

X j = 2N + 2 1¤i,j¤N

(In these statements only one half of the entries of the Pfa¬an is given, the other half

being uniquely determined by skew-symmetry).

This result looks somehow technical, but its usefulness was su¬ciently proved by

its applications in the enumeration of plane partitions and tableaux in [58] and [169,

Sec. 7].

Another technical, but useful result is due to Goulden and Jackson [61, Theorem 2.1].

Lemma 19. Let Fm (t), Gm (t) and Hm (t) by formal power series, with Hm (0) = 0,

m = 0, 1, . . . , n ’ 1. Then for any positive integer n there holds

Fj (t) Fj (t)

det CT Gi (Hj (t)) = det CT Gi (0) , (2.44)

Hj (t)i Hj (t)i

0¤i,j,¤n’1 0¤i,j¤n’1

where CT(f(t)) stands for the constant term of the Laurent series f(t).

What is the value of this theorem? In some cases, out of a given determinant eval-

uation, it immediately implies a more general one, containing (at least) one more pa-

rameter. For example, consider the determinant evaluation (3.30). Choose Fj (t) =

tj (1 + t)µ+j , Hj (t) = t2/(1 + t), and Gi (t) such that Gi (t2/(1 + t)) = (1 + t)k + (1 + t)’k

for a ¬xed k (such a choice does indeed exist; see [61, proof of Cor. 2.2]) in Lemma 19.

This yields

µ’k+i+j

µ+k+i+j µ+i+j

det + = det 2 .

2i ’ j 2i ’ j 2i ’ j

0¤i,j¤n’1 0¤i,j¤n’1

9

Another point of view, beautifully set forth in [79], is that “Pfa¬ans are more fundamental than

determinants, in the sense that determinants are merely the bipartite special case of a general sum

over matchings.”

ADVANCED DETERMINANT CALCULUS 25

Thus, out of the validity of (3.30), this enables to establish the validity of (3.32), and

even of (3.33), by choosing Fj (t) and Hj (t) as above, but Gi (t) such that Gi (t2/(1+t)) =

(1 + t)xi + (1 + t)’xi , i = 0, 1, . . . , n ’ 1.

3. A list of determinant evaluations

In this section I provide a list of determinant evaluations, some of which are very

frequently met, others maybe not so often. In any case, I believe that all of them

are useful or attractive, or even both. However, this is not intended to be, and cannot

possibly be, an exhaustive list of known determinant evaluations. The selection depends

totally on my taste. This may explain that many of these determinants arose in the

enumeration of plane partitions and rhombus tilings. On the other hand, it is exactly

this ¬eld (see [138, 148, 163, 165] for more information on these topics) which is a

particular rich source of nontrivial determinant evaluations. If you do not ¬nd “your”

determinant here, then, at least, the many references given in this section or the general

results and methods from Section 2 may turn out to be helpful.

Throughout this section we use the standard hypergeometric and basic hypergeomet-

ric notations. To wit, for nonnegative integers k the shifted factorial (a)k is de¬ned (as

already before) by

(a)k := a(a + 1) · · · (a + k ’ 1),

so that in particular (a)0 := 1. Similarly, for nonnegative integers k the shifted q-

factorial (a; q)k is given by

(a; q)k := (1 ’ a)(1 ’ aq) · · · (1 ’ aq k’1),

so that (a; q)0 := 1. Sometimes we make use of the notations [±]q := (1 ’ q ± )/(1 ’ q),

[n]q ! := [n]q [n ’ 1]q · · · [1]q , [0]q ! := 1. The q-binomial coe¬cient is de¬ned by

[±]q [± ’ 1]q · · · [± ’ k + 1]q (1 ’ q ±)(1 ’ q ±’1) · · · (1 ’ q ±’k+1 )

±

:= = .

(1 ’ q k )(1 ’ q k’1 ) · · · (1 ’ q)

k [k]q !

q

Clearly we have limq’1 [ ± ]q = ± .

k k

Occasionally shifted (q-)factorials will appear which contain a subscript which is a

negative integer. By convention, a shifted factorial (a)k , where k is a negative integer, is

interpreted as (a)k := 1/(a ’ 1)(a ’ 2) · · · (a + k), whereas a shifted q-factorial (a; q)k ,

where k is a negative integer, is interpreted as (a; q)k := 1/(1 ’ q a’1)(1 ’ q a’2 ) · · ·

(1 ’ q a+k ). (A uniform way to de¬ne the shifted factorial, for positive and negative k,

is by (a)k := “(a + k)/“(a), respectively by an appropriate limit in case that a or a + k

is a nonpositive integer, see [62, Sec. 5.5, p. 211f]. A uniform way to de¬ne the shifted

q-factorial is by means of (a; q)k := (a; q)∞/(aq k ; q)∞ , see [55, (1.2.30)].)

We begin our list with two determinant evaluations which generalize the Vander-

monde determinant evaluation (2.1) in a nonstandard way. The determinants appearing

in these evaluations can be considered as “augmentations” of the Vandermonde deter-

minant by columns which are formed by di¬erentiating “Vandermonde-type” columns.

(Thus, these determinants can also be considered as certain generalized Wronskians.)

Occurences of the ¬rst determinant can be found e.g. in [45], [107, App. A.16], [108,

(7.1.3)], [154], [187]. (It is called “con¬‚uent alternant” in [107, 108].) The motivation

in [45] to study these determinants came from Hermite interpolation and the analysis

of linear recursion relations. In [107, App. A.16], special cases of these determinants

26 C. KRATTENTHALER

are used in the context of random matrices. Special cases arose also in the context of

transcendental number theory (see [131, Sec. 4]).

Theorem 20. Let n be a nonnegative integer, and let Am (X) denote the n — m matrix

«

1 0 0 0 ... 0

¬X ·

1 0 0 ... 0

¬2 ·

¬X ·

2X 2 0 ... 0

¬3 ·,

¬X ·

2

3X 6X 6 ... 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X n’1 (n ’ 1)X n’2 (n ’ 1)(n ’ 2)X n’3 . . . . . . (n ’ 1) · · · (n ’ m + 1)X n’m

i.e., any next column is formed by di¬erentiating the previous column with respect to

X. Given a composition of n, n = m1 + · · · + m , there holds

mi ’1

(Xj ’ Xi )mi mj . (3.1)

det Am1 (X1 ) Am2 (X2 ) . . . Am (X ) = j!

1¤i,j,¤n

i=1 j=1 1¤i<j¤

The paper [45] has as well an “Abel-type” variation of this result.

Theorem 21. Let n be a nonnegative integer, and let Bm (X) denote the n — m matrix

«

1 0 0 0 ... 0

¬X ·

X X X ... X

¬2 ·

¬X ·

2 2 2 m’1 2

2X 4X 8X . . . 2 X

¬3 ·,

¬X ·

3 3 3 m’1 3

3X 9X 27X . . . 3 X

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X n’1 (n ’ 1)X n’1 (n ’ 1)2 X n’1 . . . . . . . . . . (n ’ 1)m’1 X n’1

i.e., any next column is formed by applying the operator X(d/dX). Given a composition

of n, n = m1 + · · · + m , there holds

mi ’1

(mi )

(Xj ’ Xi )mi mj .

2

det Bm1 (X1 ) Bm2 (X2 ) . . . Bm (X ) = Xi j!

1¤i,j,¤n

i=1 j=1 1¤i<j¤

(3.2)

As Alain Lascoux taught me, the natural environment for this type of determinants

is divided di¬erences and (generalized) discrete Wronskians. The divided di¬erence ‚x,y

is a linear operator which maps polynomials in x and y to polynomials symmetric in x

and y, and is de¬ned by

f(x, y) ’ f(y, x)

‚x,y f(x, y) = .

x’y

Divided di¬erences have been introduced by Newton to solve the interpolation prob-

lem in one variable. (See [100] for an excellent introduction to interpolation, divided

di¬erences, and related matters, such as Schur functions and Schubert polynomials.) In

ADVANCED DETERMINANT CALCULUS 27

fact, given a polynomial g(x) in x, whose coe¬cients do not depend on a1, a2, . . . , am ,

Newton™s interpolation formula reads as follows (cf. e.g. [100, (Ni2)]),

g(x) = g(a1 ) + (x ’ a1)‚a1 ,a2 g(a1) + (x ’ a1)(x ’ a2 )‚a2 ,a3 ‚a1 ,a2 g(a1 )

+ (x ’ a1 )(x ’ a2 )(x ’ a3 )‚a3,a4 ‚a2 ,a3 ‚a1 ,a2 g(a1) + · · · . (3.3)

Now suppose that f1 (x), f2(x), . . . , fn (x) are polynomials in one variable x, whose

coe¬cients do not depend on a1, a2, . . . , an , and consider the determinant

det (fi (aj )). (3.4)

1¤i,j,¤n

Let us for the moment concentrate on the ¬rst m1 columns of this determinant. We

may apply (3.3), and write

fi (aj ) = fi (a1) + (aj ’ a1)‚a1 ,a2 fi (a1) + (aj ’ a1)(aj ’ a2)‚a2 ,a3 ‚a1 ,a2 fi (a1)

+ · · · + (aj ’ a1)(aj ’ a2) · · · (aj ’ aj’1 )‚aj’1 ,aj · · · ‚a2 ,a3 ‚a1,a2 fi (a1 ),

j = 1, 2, . . . , m1. Following [100, Proof of Lemma (Ni5)], we may perform column

reductions to the e¬ect that the determinant (3.4), with column j replaced by

(aj ’ a1 )(aj ’ a2 ) · · · (aj ’ aj’1 )‚aj’1 ,aj · · · ‚a2,a3 ‚a1 ,a2 fi (a1),

j = 1, 2, . . . , m1, has the same value as the original determinant. Clearly, the product

k=1 (aj ’ ak ) can be taken out of column j, j = 1, 2, . . . , m1 . Similar reductions can

j’1

be applied to the next m2 columns, then to the next m3 columns, etc.

This proves the following fact about generalized discrete Wronskians:

Lemma 22. Let n be a nonnegative integer, and let Wm (x1 , x2, . . . , xm) denote the

n — m matrix ‚xj’1 ,xj · · · ‚x2 ,x3 ‚x1 ,x2 fi (x1) 1¤i¤n, 1¤j¤m . Given a composition of n,

n = m1 + · · · + m , there holds

det Wm1 (a1, . . . , am1 ) Wm2 (am1 +1 , . . . , am1 +m2 ) . . . Wm (am1 +···+m ’1 +1 , . . . , an )

1¤i,j,¤n

(aj ’ ai ) . (3.5)

= det (fi (aj ))

1¤i,j,¤n

m1 +···+mk’1 +1¤i<j¤m1 +···+mk

k=1

If we now choose fi (x) := xi’1 , so that det1¤i,j,¤n (fi (aj )) is a Vandermonde deter-

minant, then the right-hand side of (3.5) factors completely by (2.1). The ¬nal step

to obtain Theorem 20 is to let a1 ’ X1 , a2 ’ X1 , . . . , am1 ’ X1 , am1 +1 ’ X2 , . . . ,

am1 +m2 ’ X2 , etc., in (3.5). This does indeed yield (3.1), because

j’1

1 d

lim . . . lim lim ‚xj’1 ,xj · · · ‚x2 ,x3 ‚x1 ,x2 g(x1) = g(x),

(j ’ 1)! dx

xj ’x x2 ’x x1 ’x

as is easily veri¬ed.

The Abel-type variation in Theorem 21 follows from Theorem 20 by multiplying

column j in (3.1) by X1 for j = 1, 2, . . . , m1, by X2 1 ’1 for j = m1 +1, m1 +2, . . . , m2 ,

j’1 j’m

etc., and by then using the relation

d d

Xg(X) ’ g(X)

X g(X) =

dX dX

28 C. KRATTENTHALER

j’1 i’1

many times, so that a typical entry Xk (d/dXk )j’1 Xk in row i and column j of the

i’1

k-th submatrix is expressed as (Xk (d/dXk ))j’1 Xk plus a linear combination of terms

(Xk (d/dXk ))s Xk with s < j ’ 1. Simple column reductions then yield (3.2).

i’1

It is now not very di¬cult to adapt this analysis to derive, for example, q-analogues

of Theorems 20 and 21. The results below do actually contain q-analogues of extensions

of Theorems 20 and 21.

Theorem 23. Let n be a nonnegative integer, and let Am (X) denote the n — m matrix

«

[C]q X ’1 [C]q [C ’ 1]q X ’2

1

¬X [C + 1]q [C]q X ’1

[C + 1]q

¬2

¬X [C + 2]q X [C + 2]q [C + 1]q

¬

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X n’1 [C + n ’ 1]q X n’2 [C + n ’ 1]q [C + n ’ 2]q X n’3

[C]q · · · [C ’ m + 2]q X 1’m

...

·

[C + 1]q · · · [C ’ m + 3]q X 2’m

... ·

·,

[C + 2]q · · · [C ’ m + 4]q X 3’m

... ·

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . [C + n ’ 1]q · · · [C + n ’ m + 1]q X n’m

i.e., any next column is formed by applying the operator X ’C Dq X C , with Dq denoting

the usual q-derivative, Dq f(X) := (f(qX) ’ f(X))/(q ’ 1)X. Given a composition of

n, n = m1 + · · · + m , there holds

det Am1 (X1 ) Am2 (X2 ) . . . Am (X )

1¤i,j,¤n

mi ’1 mj ’1

mi ’1

(q t’s Xj ’ Xi ), (3.6)

= q N1 [j]q !

i=1 j=1 s=0 t=0

1¤i<j¤

where N1 is the quantity

mi

(C + j + m1 + · · · + mi’1 ’ 1)(mi ’ j) ’ ’ ’ mj

mi mj mi

mi .

3 2 2

i=1 j=1 1¤i<j¤

To derive (3.6) one would choose strings of geometric sequences for the variables aj