ADVANCED DETERMINANT CALCULUS
C. KRATTENTHALER†
Institut f ü r Mathematik der Universität Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
E-mail:
WWW: http://radon.mat.univie.ac.at/People/kratt
Dedicated to the pioneer of determinant evaluations (among many other things),
George Andrews
ABSTRACT. The purpose of this article is threefold. First, it provides the reader with
a few useful and efficient tools which should enable her/him to evaluate nontrivial de-
terminants for the case such a determinant should appear in her/his research. Second,
it lists a number of such determinants that have been already evaluated, together with
explanations which tell in which contexts they have appeared. Third, it points out
references where further such determinant evaluations can be found.
1. Introduction
Imagine, you are working on a problem. As things develop it turns out that, in
order to solve your problem, you need to evaluate a certain determinant. Maybe your
determinant is
1
det , (1.1)
1≤i,j,≤n i+j
or
a+b
det , (1.2)
1≤i,j≤n a−i+j
or it is possibly
µ+i+j
det , (1.3)
0≤i,j≤n−1 2i − j
1991 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 05A17 05A18
05A30 05E10 05E15 11B68 11B73 11C20 15A15 33C45 33D45.
Key words and phrases. Determinants, Vandermonde determinant, Cauchy’s double alternant,
Pfaffian, discrete Wronskian, Hankel determinants, orthogonal polynomials, Chebyshev polynomials,
Meixner polynomials, Meixner–Pollaczek polynomials, Hermite polynomials, Charlier polynomials, La-
guerre polynomials, Legendre polynomials, ultraspherical polynomials, continuous Hahn polynomials,
continued fractions, binomial coefficient, Genocchi numbers, Bernoulli numbers, Stirling numbers, Bell
numbers, Euler numbers, divided difference, interpolation, plane partitions, tableaux, rhombus tilings,
lozenge tilings, alternating sign matrices, noncrossing partitions, perfect matchings, permutations,
inversion number, major index, descent algebra, noncommutative symmetric functions.
† Research partially supported by the Austrian Science Foundation FWF, grants P12094-MAT and
P13190-MAT.
1
,2 C. KRATTENTHALER
or maybe
det x+y+j x+y+j
− . (1.4)
1≤i,j≤n x − i + 2j x + i + 2j
Honestly, which ideas would you have? (Just to tell you that I do not ask for something
impossible: Each of these four determinants can be evaluated in “closed form”. If you
want to see the solutions immediately, plus information where these determinants come
from, then go to (2.7), (2.17)/(3.12), (2.19)/(3.30), respectively (3.47).)
Okay, let us try some row and column manipulations. Indeed, although it is not
completely trivial (actually, it is quite a challenge), that would work for the first two
determinants, (1.1) and (1.2), although I do not recommend that. However, I do not
recommend at all that you try this with the latter two determinants, (1.3) and (1.4). I
promise that you will fail. (The determinant (1.3) does not look much more complicated
than (1.2). Yet, it is.)
So, what should we do instead?
Of course, let us look in the literature! Excellent idea. We may have the problem
of not knowing where to start looking. Good starting points are certainly classics like
[119], [120], [121], [127] and [178] 1. This will lead to the first success, as (1.1) does
indeed turn up there (see [119, vol. III, p. 311]). Yes, you will also find evaluations for
(1.2) (see e.g. [126]) and (1.3) (see [112, Theorem 7]) in the existing literature. But at
the time of the writing you will not, to the best of my knowledge, find an evaluation of
(1.4) in the literature.
The purpose of this article is threefold. First, I want to describe a few useful and
efficient tools which should enable you to evaluate nontrivial determinants (see Sec-
tion 2). Second, I provide a list containing a number of such determinants that have
been already evaluated, together with explanations which tell in which contexts they
have appeared (see Section 3). Third, even if you should not find your determinant
in this list, I point out references where further such determinant evaluations can be
found, maybe your determinant is there.
Most important of all is that I want to convince you that, today,
Evaluating determinants is not (okay: may not be) difficult!
When George Andrews, who must be rightly called the pioneer of determinant evalua-
tions, in the seventies astounded the combinatorial community by his highly nontrivial
determinant evaluations (solving difficult enumeration problems on plane partitions),
it was really difficult. His method (see Section 2.6 for a description) required a good
“guesser” and an excellent “hypergeometer” (both of which he was and is). While at
that time especially to be the latter was quite a task, in the meantime both guessing and
evaluating binomial and hypergeometric sums has been largely trivialized, as both can
be done (most of the time) completely automatically. For guessing (see Appendix A)
1Turnbull’s
book [178] does in fact contain rather lots of very general identities satisfied by determi-
nants, than determinant “evaluations” in the strict sense of the word. However, suitable specializations
of these general identities do also yield “genuine” evaluations, see for example Appendix B. Since the
value of this book may not be easy to appreciate because of heavy notation, we refer the reader to
[102] for a clarification of the notation and a clear presentation of many such identities.
, ADVANCED DETERMINANT CALCULUS 3
this is due to tools like Superseeker2, gfun and Mgfun3 [152, 24], and Rate4 (which is
by far the most primitive of the three, but it is the most effective in this context). For
“hypergeometrics” this is due to the “WZ-machinery”5 (see [130, 190, 194, 195, 196]).
And even if you should meet a case where the WZ-machinery should exhaust your com-
puter’s capacity, then there are still computer algebra packages like HYP and HYPQ 6,
or HYPERG7, which make you an expert hypergeometer, as these packages comprise
large parts of the present hypergeometric knowledge, and, thus, enable you to con-
veniently manipulate binomial and hypergeometric series (which George Andrews did
largely by hand) on the computer. Moreover, as of today, there are a few new (perhaps
just overlooked) insights which make life easier in many cases. It is these which form
large parts of Section 2.
So, if you see a determinant, don’t be frightened, evaluate it yourself!
2. Methods for the evaluation of determinants
In this section I describe a few useful methods and theorems which (may) help you
to evaluate a determinant. As was mentioned already in the Introduction, it is always
possible that simple-minded things like doing some row and/or column operations, or
applying Laplace expansion may produce an (usually inductive) evaluation of a deter-
minant. Therefore, you are of course advised to try such things first. What I am
mainly addressing here, though, is the case where that first, “simple-minded” attempt
failed. (Clearly, there is no point in addressing row and column operations, or Laplace
expansion.)
Yet, we must of course start (in Section 2.1) with some standard determinants, such
as the Vandermonde determinant or Cauchy’s double alternant. These are of course
well-known.
In Section 2.2 we continue with some general determinant evaluations that generalize
the evaluation of the Vandermonde determinant, which are however apparently not
equally well-known, although they should be. In fact, I claim that about 80 % of the
determinants that you meet in “real life,” and which can apparently be evaluated, are a
special case of just the very first of these (Lemma 3; see in particular Theorem 26 and
the subsequent remarks). Moreover, as is demonstrated in Section 2.2, it is pure routine
to check whether a determinant is a special case of one of these general determinants.
Thus, it can be really considered as a “method” to see if a determinant can be evaluated
by one of the theorems in Section 2.2.
2the
electronic version of the “Encyclopedia of Integer Sequences” [162, 161], written and developed
by Neil Sloane and Simon Plouffe; see http://www.research.att.com/~njas/sequences/ol.html
3written by Bruno Salvy and Paul Zimmermann, respectively Frederic Chyzak; available from
http://pauillac.inria.fr/algo/libraries/libraries.html
4written in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt;
the Maple equivalent GUESS by Franc¸ois B´eraud and Bruno Gauthier is available from
http://www-igm.univ-mlv.fr/~gauthier
5Maple
implementations written by Doron Zeilberger are available from
http://www.math.temple.edu/~zeilberg, Mathematica implementations written by
Peter Paule, Axel Riese, Markus Schorn, Kurt Wegschaider are available from
http://www.risc.uni-linz.ac.at/research/combinat/risc/software
6written in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt
7written in Maple by Bruno Ghauthier; available from http://www-igm.univ-mlv.fr/~gauthier
, 4 C. KRATTENTHALER
The next method which I describe is the so-called “condensation method” (see Sec-
tion 2.3), a method which allows to evaluate a determinant inductively (if the method
works).
In Section 2.4, a method, which I call the “identification of factors” method, is de-
scribed. This method has been extremely successful recently. It is based on a very
simple idea, which comes from one of the standard proofs of the Vandermonde deter-
minant evaluation (which is therefore described in Section 2.1).
The subject of Section 2.5 is a method which is based on finding one or more differen-
tial or difference equations for the matrix of which the determinant is to be evaluated.
Section 2.6 contains a short description of George Andrews’ favourite method, which
basically consists of explicitly doing the LU-factorization of the matrix of which the
determinant is to be evaluated.
The remaining subsections in this section are conceived as a complement to the pre-
ceding. In Section 2.7 a special type of determinants is addressed, Hankel determinants.
(These are determinants of the form det1≤i,j≤n(ai+j), and are sometimes also called per-
symmetric or Turá nian determinants.) As is explained there, you should expect that a
Hankel determinant evaluation is to be found in the domain of orthogonal polynomials
and continued fractions. Eventually, in Section 2.8 a few further, possibly useful results
are exhibited.
Before we finally move into the subject, it must be pointed out that the methods
of determinant evaluation as presented here are ordered according to the conditions a
determinant must satisfy so that the method can be applied to it, from “stringent” to
“less stringent”. I. e., first come the methods which require that the matrix of which
the determinant is to be taken satisfies a lot of conditions (usually: it contains a lot of
parameters, at least, implicitly), and in the end comes the method (LU-factorization)
which requires nothing. In fact, this order (of methods) is also the order in which I
recommend that you try them on your determinant. That is, what I suggest is (and
this is the rule I follow):
(0) First try some simple-minded things (row and column operations, Laplace expan-
sion). Do not waste too much time. If you encounter a Hankel-determinant then
see Section 2.7.
(1) If that fails, check whether your determinant is a special case of one of the general
determinants in Sections 2.2 (and 2.1).
(2) If that fails, see if the condensation method (see Section 2.3) works. (If necessary,
try to introduce more parameters into your determinant.)
(3) If that fails, try the “identification of factors” method (see Section 2.4). Alterna-
tively, and in particular if your matrix of which you want to find the determinant
is the matrix defining a system of differential or difference equations, try the dif-
ferential/difference equation method of Section 2.5. (If necessary, try to introduce
a parameter into your determinant.)
(4) If that fails, try to work out the LU-factorization of your determinant (see Sec-
tion 2.6).
(5) If all that fails, then we are really in trouble. Perhaps you have to put more efforts
into determinant manipulations (see suggestion (0))? Sometimes it is worthwile
to interpret the matrix whose determinant you want to know as a linear map and
try to find a basis on which this map acts triangularly, or even diagonally (this
C. KRATTENTHALER†
Institut f ü r Mathematik der Universität Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
E-mail:
WWW: http://radon.mat.univie.ac.at/People/kratt
Dedicated to the pioneer of determinant evaluations (among many other things),
George Andrews
ABSTRACT. The purpose of this article is threefold. First, it provides the reader with
a few useful and efficient tools which should enable her/him to evaluate nontrivial de-
terminants for the case such a determinant should appear in her/his research. Second,
it lists a number of such determinants that have been already evaluated, together with
explanations which tell in which contexts they have appeared. Third, it points out
references where further such determinant evaluations can be found.
1. Introduction
Imagine, you are working on a problem. As things develop it turns out that, in
order to solve your problem, you need to evaluate a certain determinant. Maybe your
determinant is
1
det , (1.1)
1≤i,j,≤n i+j
or
a+b
det , (1.2)
1≤i,j≤n a−i+j
or it is possibly
µ+i+j
det , (1.3)
0≤i,j≤n−1 2i − j
1991 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 05A17 05A18
05A30 05E10 05E15 11B68 11B73 11C20 15A15 33C45 33D45.
Key words and phrases. Determinants, Vandermonde determinant, Cauchy’s double alternant,
Pfaffian, discrete Wronskian, Hankel determinants, orthogonal polynomials, Chebyshev polynomials,
Meixner polynomials, Meixner–Pollaczek polynomials, Hermite polynomials, Charlier polynomials, La-
guerre polynomials, Legendre polynomials, ultraspherical polynomials, continuous Hahn polynomials,
continued fractions, binomial coefficient, Genocchi numbers, Bernoulli numbers, Stirling numbers, Bell
numbers, Euler numbers, divided difference, interpolation, plane partitions, tableaux, rhombus tilings,
lozenge tilings, alternating sign matrices, noncrossing partitions, perfect matchings, permutations,
inversion number, major index, descent algebra, noncommutative symmetric functions.
† Research partially supported by the Austrian Science Foundation FWF, grants P12094-MAT and
P13190-MAT.
1
,2 C. KRATTENTHALER
or maybe
det x+y+j x+y+j
− . (1.4)
1≤i,j≤n x − i + 2j x + i + 2j
Honestly, which ideas would you have? (Just to tell you that I do not ask for something
impossible: Each of these four determinants can be evaluated in “closed form”. If you
want to see the solutions immediately, plus information where these determinants come
from, then go to (2.7), (2.17)/(3.12), (2.19)/(3.30), respectively (3.47).)
Okay, let us try some row and column manipulations. Indeed, although it is not
completely trivial (actually, it is quite a challenge), that would work for the first two
determinants, (1.1) and (1.2), although I do not recommend that. However, I do not
recommend at all that you try this with the latter two determinants, (1.3) and (1.4). I
promise that you will fail. (The determinant (1.3) does not look much more complicated
than (1.2). Yet, it is.)
So, what should we do instead?
Of course, let us look in the literature! Excellent idea. We may have the problem
of not knowing where to start looking. Good starting points are certainly classics like
[119], [120], [121], [127] and [178] 1. This will lead to the first success, as (1.1) does
indeed turn up there (see [119, vol. III, p. 311]). Yes, you will also find evaluations for
(1.2) (see e.g. [126]) and (1.3) (see [112, Theorem 7]) in the existing literature. But at
the time of the writing you will not, to the best of my knowledge, find an evaluation of
(1.4) in the literature.
The purpose of this article is threefold. First, I want to describe a few useful and
efficient tools which should enable you to evaluate nontrivial determinants (see Sec-
tion 2). Second, I provide a list containing a number of such determinants that have
been already evaluated, together with explanations which tell in which contexts they
have appeared (see Section 3). Third, even if you should not find your determinant
in this list, I point out references where further such determinant evaluations can be
found, maybe your determinant is there.
Most important of all is that I want to convince you that, today,
Evaluating determinants is not (okay: may not be) difficult!
When George Andrews, who must be rightly called the pioneer of determinant evalua-
tions, in the seventies astounded the combinatorial community by his highly nontrivial
determinant evaluations (solving difficult enumeration problems on plane partitions),
it was really difficult. His method (see Section 2.6 for a description) required a good
“guesser” and an excellent “hypergeometer” (both of which he was and is). While at
that time especially to be the latter was quite a task, in the meantime both guessing and
evaluating binomial and hypergeometric sums has been largely trivialized, as both can
be done (most of the time) completely automatically. For guessing (see Appendix A)
1Turnbull’s
book [178] does in fact contain rather lots of very general identities satisfied by determi-
nants, than determinant “evaluations” in the strict sense of the word. However, suitable specializations
of these general identities do also yield “genuine” evaluations, see for example Appendix B. Since the
value of this book may not be easy to appreciate because of heavy notation, we refer the reader to
[102] for a clarification of the notation and a clear presentation of many such identities.
, ADVANCED DETERMINANT CALCULUS 3
this is due to tools like Superseeker2, gfun and Mgfun3 [152, 24], and Rate4 (which is
by far the most primitive of the three, but it is the most effective in this context). For
“hypergeometrics” this is due to the “WZ-machinery”5 (see [130, 190, 194, 195, 196]).
And even if you should meet a case where the WZ-machinery should exhaust your com-
puter’s capacity, then there are still computer algebra packages like HYP and HYPQ 6,
or HYPERG7, which make you an expert hypergeometer, as these packages comprise
large parts of the present hypergeometric knowledge, and, thus, enable you to con-
veniently manipulate binomial and hypergeometric series (which George Andrews did
largely by hand) on the computer. Moreover, as of today, there are a few new (perhaps
just overlooked) insights which make life easier in many cases. It is these which form
large parts of Section 2.
So, if you see a determinant, don’t be frightened, evaluate it yourself!
2. Methods for the evaluation of determinants
In this section I describe a few useful methods and theorems which (may) help you
to evaluate a determinant. As was mentioned already in the Introduction, it is always
possible that simple-minded things like doing some row and/or column operations, or
applying Laplace expansion may produce an (usually inductive) evaluation of a deter-
minant. Therefore, you are of course advised to try such things first. What I am
mainly addressing here, though, is the case where that first, “simple-minded” attempt
failed. (Clearly, there is no point in addressing row and column operations, or Laplace
expansion.)
Yet, we must of course start (in Section 2.1) with some standard determinants, such
as the Vandermonde determinant or Cauchy’s double alternant. These are of course
well-known.
In Section 2.2 we continue with some general determinant evaluations that generalize
the evaluation of the Vandermonde determinant, which are however apparently not
equally well-known, although they should be. In fact, I claim that about 80 % of the
determinants that you meet in “real life,” and which can apparently be evaluated, are a
special case of just the very first of these (Lemma 3; see in particular Theorem 26 and
the subsequent remarks). Moreover, as is demonstrated in Section 2.2, it is pure routine
to check whether a determinant is a special case of one of these general determinants.
Thus, it can be really considered as a “method” to see if a determinant can be evaluated
by one of the theorems in Section 2.2.
2the
electronic version of the “Encyclopedia of Integer Sequences” [162, 161], written and developed
by Neil Sloane and Simon Plouffe; see http://www.research.att.com/~njas/sequences/ol.html
3written by Bruno Salvy and Paul Zimmermann, respectively Frederic Chyzak; available from
http://pauillac.inria.fr/algo/libraries/libraries.html
4written in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt;
the Maple equivalent GUESS by Franc¸ois B´eraud and Bruno Gauthier is available from
http://www-igm.univ-mlv.fr/~gauthier
5Maple
implementations written by Doron Zeilberger are available from
http://www.math.temple.edu/~zeilberg, Mathematica implementations written by
Peter Paule, Axel Riese, Markus Schorn, Kurt Wegschaider are available from
http://www.risc.uni-linz.ac.at/research/combinat/risc/software
6written in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt
7written in Maple by Bruno Ghauthier; available from http://www-igm.univ-mlv.fr/~gauthier
, 4 C. KRATTENTHALER
The next method which I describe is the so-called “condensation method” (see Sec-
tion 2.3), a method which allows to evaluate a determinant inductively (if the method
works).
In Section 2.4, a method, which I call the “identification of factors” method, is de-
scribed. This method has been extremely successful recently. It is based on a very
simple idea, which comes from one of the standard proofs of the Vandermonde deter-
minant evaluation (which is therefore described in Section 2.1).
The subject of Section 2.5 is a method which is based on finding one or more differen-
tial or difference equations for the matrix of which the determinant is to be evaluated.
Section 2.6 contains a short description of George Andrews’ favourite method, which
basically consists of explicitly doing the LU-factorization of the matrix of which the
determinant is to be evaluated.
The remaining subsections in this section are conceived as a complement to the pre-
ceding. In Section 2.7 a special type of determinants is addressed, Hankel determinants.
(These are determinants of the form det1≤i,j≤n(ai+j), and are sometimes also called per-
symmetric or Turá nian determinants.) As is explained there, you should expect that a
Hankel determinant evaluation is to be found in the domain of orthogonal polynomials
and continued fractions. Eventually, in Section 2.8 a few further, possibly useful results
are exhibited.
Before we finally move into the subject, it must be pointed out that the methods
of determinant evaluation as presented here are ordered according to the conditions a
determinant must satisfy so that the method can be applied to it, from “stringent” to
“less stringent”. I. e., first come the methods which require that the matrix of which
the determinant is to be taken satisfies a lot of conditions (usually: it contains a lot of
parameters, at least, implicitly), and in the end comes the method (LU-factorization)
which requires nothing. In fact, this order (of methods) is also the order in which I
recommend that you try them on your determinant. That is, what I suggest is (and
this is the rule I follow):
(0) First try some simple-minded things (row and column operations, Laplace expan-
sion). Do not waste too much time. If you encounter a Hankel-determinant then
see Section 2.7.
(1) If that fails, check whether your determinant is a special case of one of the general
determinants in Sections 2.2 (and 2.1).
(2) If that fails, see if the condensation method (see Section 2.3) works. (If necessary,
try to introduce more parameters into your determinant.)
(3) If that fails, try the “identification of factors” method (see Section 2.4). Alterna-
tively, and in particular if your matrix of which you want to find the determinant
is the matrix defining a system of differential or difference equations, try the dif-
ferential/difference equation method of Section 2.5. (If necessary, try to introduce
a parameter into your determinant.)
(4) If that fails, try to work out the LU-factorization of your determinant (see Sec-
tion 2.6).
(5) If all that fails, then we are really in trouble. Perhaps you have to put more efforts
into determinant manipulations (see suggestion (0))? Sometimes it is worthwile
to interpret the matrix whose determinant you want to know as a linear map and
try to find a basis on which this map acts triangularly, or even diagonally (this
