paginas.matem.unam.mx · Introduction: Tur an, Karlin and Szeg}o Determinants whose entries are...

Invariant properties for Wronskian typedeterminants of classical and classical discrete

orthogonal polynomials

Antonio J. DuranUniversidad de Sevilla

Honoring Prof. Natig Atakishiyev

Antonio J. Duran Universidad de Sevilla Wronskian type determinants

Introduction: Turan, Karlin and Szego

Determinants whose entries are orthogonal polynomials is a long studiedsubject.Turan inequality (1948): if pn, n ≥ 0, are the Legendre polynomials, then∣∣∣∣ pn(x) pn+1(x)

pn+1(x) pn+2(x)

∣∣∣∣ < 0, −1 < x < 1

Karlin and Szego’s generalization for Hankel determinants (1961): if k is eventhen ∣∣∣∣∣∣∣∣∣

pn(x) pn+1(x) · · · pn+k−1(x)pn+1(x) pn+2(x) · · · pn+k(x)

......

. . ....

pn+k−1(x) pn+k(x) · · · pn+2k(x)

∣∣∣∣∣∣∣∣∣has constant sign for

1 pn=Gegenbauer polynomials and −1 < x < 1.

2 pn=Laguerre polynomials and 0 < x .

3 pn=Hermite polynomials and x ∈ R.



Determinants whose entries are orthogonal polynomials is a long studiedsubject.

Turan inequality (1948): if pn, n ≥ 0, are the Legendre polynomials, then∣∣∣∣ pn(x) pn+1(x)pn+1(x) pn+2(x)

∣∣∣∣ < 0, −1 < x < 1



......

. . ....

pn+k−1(x) pn+k(x) · · · pn+2k(x)







Determinants whose entries are orthogonal polynomials is a long studiedsubject.Turan inequality (1948): if pn, n ≥ 0, are the Legendre polynomials, then∣∣∣∣ pn(x) pn+1(x)

pn+1(x) pn+2(x)

∣∣∣∣ < 0, −1 < x < 1



......

. . ....

pn+k−1(x) pn+k(x) · · · pn+2k(x)







Karlin and Szego also studied Casorati determinants of orthogonal polynomialswith respect to a discrete measure∣∣∣∣∣∣∣∣∣

pm(x) pm(x + 1) · · · pm(x + n − 1)pm+1(x) pm+1(x + 1) · · · pm+1(x + n − 1)

......

. . ....

pm+n−1(x) pm+n−1(x + 1) · · · pm+n−1(x + n − 1)

∣∣∣∣∣∣∣∣∣In particular they studied the sign of these determinants when pn are theCharlier, Meixner, Krawtchouk and Chebyshev polynomials.However, they did not notice certain surprising symmetries and invariantproperties these Casorati determinants enjoy.





......

. . ....

pm+n−1(x) pm+n−1(x + 1) · · · pm+n−1(x + n − 1)

∣∣∣∣∣∣∣∣∣

In particular they studied the sign of these determinants when pn are theCharlier, Meixner, Krawtchouk and Chebyshev polynomials.However, they did not notice certain surprising symmetries and invariantproperties these Casorati determinants enjoy.





......

. . ....

pm+n−1(x) pm+n−1(x + 1) · · · pm+n−1(x + n − 1)

∣∣∣∣∣∣∣∣∣In particular they studied the sign of these determinants when pn are theCharlier, Meixner, Krawtchouk and Chebyshev polynomials.

However, they did not notice certain surprising symmetries and invariantproperties these Casorati determinants enjoy.





......

. . ....

pm+n−1(x) pm+n−1(x + 1) · · · pm+n−1(x + n − 1)

∣∣∣∣∣∣∣∣∣In particular they studied the sign of these determinants when pn are theCharlier, Meixner, Krawtchouk and Chebyshev polynomials.However, they did not notice certain surprising symmetries and invariantproperties these Casorati determinants enjoy.


Symmetries for Casoratian and Wroskian determinants

Here it is one of these symmetries.can ≡ Charlier polynomials (normalization can(x) = xn/n! + · · · )F = {f1, · · · , fk} positive integers (written in increasing size)

CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1

Υ : the set formed by all finite sets of positive integers

I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.

#F = k then #I (F ) = m = maxF − k + 1Then (A.J.D. 2016)

CaF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)

HF ,x = iwF HI (F ),−ix

The purpose of this talk is to show an approach to these symmetries usingKrall discrete orthogonal polynomials.We also show that these symmetries are related to the equivalence betweeneigenstate adding and deleting Darboux transformations for solvable quantummechanical systems, and with Selberg type integrals and constant termidentities of certain multivariate Laurent expansions.



Here it is one of these symmetries.

can ≡ Charlier polynomials (normalization can(x) = xn/n! + · · · )F = {f1, · · · , fk} positive integers (written in increasing size)

CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.


CaF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)





Here it is one of these symmetries.can ≡ Charlier polynomials (normalization can(x) = xn/n! + · · · )

F = {f1, · · · , fk} positive integers (written in increasing size)

CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.


CaF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)






CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.


CaF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)






CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.I ({1, 2, · · · , k}) = {k}, I ({1, k}) = {1, 2, · · · , k − 2, k}


CaF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)






CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.

#F = k then #I (F ) = m = maxF − k + 1

Then (A.J.D. 2016)

CaF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)






CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.

#F = k then #I (F ) = m = maxF − k + 1Then (A.J.D. 2016) C

aF ,x︸︷︷︸

size k × k

= (−1)wF

C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)






CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.


aF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)






CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.


aF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)


The purpose of this talk is to show an approach to these symmetries usingKrall discrete orthogonal polynomials.

We also show that these symmetries are related to the equivalence betweeneigenstate adding and deleting Darboux transformations for solvable quantummechanical systems, and with Selberg type integrals and constant termidentities of certain multivariate Laurent expansions.




CaF ,x = det

(cafi (x + j − 1)

)ki,j=1

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


I : Υ→ Υ

I (F ) = {1, 2, · · · ,maxF} \ {maxF − f , f ∈ F}.


aF ,x︸︷︷︸

size k × k

= (−1)wF C−aI (F ),−x︸︷︷︸

size m ×m

wF =∑

F f −(k2

)




Krall orthogonal polynomials

Krall (or bispectral) polynomials (1940).They are a generalization of classical and classical discrete orthogonalpolynomials.Definition: Orthogonal polynomials that are also eigenfunctions of a differentialor difference operator of order k, k > 2.The first examples were introduced by H. Krall in 1940. Krall found all familiesof orthogonal polynomials which are common eigenfunctions of a fourth orderdifferential operator.

Differential operatorsHow can one construct Krall polynomials?Take the Laguerre (xαe−x , (0,+∞)) or Jacobi weights ((1− x)α(1 + x)β ,(−1, 1)), assume one or two of the parameters are nonnegative integers andadd a Dirac delta at one or two of the end points of the interval oforthogonality. We do not know any example of Krall polynomials associated tothe Hermite polynomials.Since the eighties a lot of effort has been devoted to this issue (withcontributions by L.L. Littlejohn, A.M. Krall, J. and R. Koekoek. A. Grunbaumand L. Haine (and collaborators), K.H. Kwon (and collaborators), A. Zhedanov,P. Iliev, A.J.D. and many others).



Krall (or bispectral) polynomials (1940).

They are a generalization of classical and classical discrete orthogonalpolynomials.Definition: Orthogonal polynomials that are also eigenfunctions of a differentialor difference operator of order k, k > 2.The first examples were introduced by H. Krall in 1940. Krall found all familiesof orthogonal polynomials which are common eigenfunctions of a fourth orderdifferential operator.




Krall (or bispectral) polynomials (1940).They are a generalization of classical and classical discrete orthogonalpolynomials.

Definition: Orthogonal polynomials that are also eigenfunctions of a differentialor difference operator of order k, k > 2.The first examples were introduced by H. Krall in 1940. Krall found all familiesof orthogonal polynomials which are common eigenfunctions of a fourth orderdifferential operator.




Krall (or bispectral) polynomials (1940).They are a generalization of classical and classical discrete orthogonalpolynomials.Definition:

Orthogonal polynomials that are also eigenfunctions of a differentialor difference operator of order k, k > 2.The first examples were introduced by H. Krall in 1940. Krall found all familiesof orthogonal polynomials which are common eigenfunctions of a fourth orderdifferential operator.




Krall (or bispectral) polynomials (1940).They are a generalization of classical and classical discrete orthogonalpolynomials.Definition: Orthogonal polynomials that are also eigenfunctions of a differentialor difference operator

of order k, k > 2.The first examples were introduced by H. Krall in 1940. Krall found all familiesof orthogonal polynomials which are common eigenfunctions of a fourth orderdifferential operator.




Krall (or bispectral) polynomials (1940).They are a generalization of classical and classical discrete orthogonalpolynomials.Definition: Orthogonal polynomials that are also eigenfunctions of a differentialor difference operator of order k, k > 2.

The first examples were introduced by H. Krall in 1940. Krall found all familiesof orthogonal polynomials which are common eigenfunctions of a fourth orderdifferential operator.





Differential operators

How can one construct Krall polynomials?Take the Laguerre (xαe−x , (0,+∞)) or Jacobi weights ((1− x)α(1 + x)β ,(−1, 1)), assume one or two of the parameters are nonnegative integers andadd a Dirac delta at one or two of the end points of the interval oforthogonality. We do not know any example of Krall polynomials associated tothe Hermite polynomials.Since the eighties a lot of effort has been devoted to this issue (withcontributions by L.L. Littlejohn, A.M. Krall, J. and R. Koekoek. A. Grunbaumand L. Haine (and collaborators), K.H. Kwon (and collaborators), A. Zhedanov,P. Iliev, A.J.D. and many others).




Differential operatorsHow can one construct Krall polynomials?

Take the Laguerre (xαe−x , (0,+∞)) or Jacobi weights ((1− x)α(1 + x)β ,(−1, 1)), assume one or two of the parameters are nonnegative integers andadd a Dirac delta at one or two of the end points of the interval oforthogonality. We do not know any example of Krall polynomials associated tothe Hermite polynomials.Since the eighties a lot of effort has been devoted to this issue (withcontributions by L.L. Littlejohn, A.M. Krall, J. and R. Koekoek. A. Grunbaumand L. Haine (and collaborators), K.H. Kwon (and collaborators), A. Zhedanov,P. Iliev, A.J.D. and many others).




Differential operatorsHow can one construct Krall polynomials?Take the Laguerre (xαe−x , (0,+∞)) or Jacobi weights ((1− x)α(1 + x)β ,(−1, 1)), assume one or two of the parameters are nonnegative integers andadd a Dirac delta at one or two of the end points of the interval oforthogonality.

We do not know any example of Krall polynomials associated tothe Hermite polynomials.Since the eighties a lot of effort has been devoted to this issue (withcontributions by L.L. Littlejohn, A.M. Krall, J. and R. Koekoek. A. Grunbaumand L. Haine (and collaborators), K.H. Kwon (and collaborators), A. Zhedanov,P. Iliev, A.J.D. and many others).




Differential operatorsHow can one construct Krall polynomials?Take the Laguerre (xαe−x , (0,+∞)) or Jacobi weights ((1− x)α(1 + x)β ,(−1, 1)), assume one or two of the parameters are nonnegative integers andadd a Dirac delta at one or two of the end points of the interval oforthogonality. We do not know any example of Krall polynomials associated tothe Hermite polynomials.

Since the eighties a lot of effort has been devoted to this issue (withcontributions by L.L. Littlejohn, A.M. Krall, J. and R. Koekoek. A. Grunbaumand L. Haine (and collaborators), K.H. Kwon (and collaborators), A. Zhedanov,P. Iliev, A.J.D. and many others).


Krall discrete polynomials.

Difference operators

The problem of finding Krall discrete polynomials was open during manydecades.What we know for the differential case has seemed to be of little help becauseadding Dirac deltas to the classical discrete measures seems not to work.Indeed, Richard Askey in 1991 explicitly posed the problem of finding the firstexamples of Krall-discrete polynomials. He suggested to study measures whichconsist of some classical discrete weights together with a Dirac delta at the endpoint(s) of the interval of orthogonality. Three years later, Bavinck, vanHaeringen and Koekoek gave a negative answer to Askey’s question: theyadded a Dirac delta at zero to the Charlier and Meixner weights andconstructed difference operators with respect to which the correspondingorthogonal polynomials are eigenfunctions... but these difference operators havealways infinite order.In 2012, this speaker found the first examples of measures whose orthogonalpolynomials are also eigenfunctions of a higher order difference operator.




The problem of finding Krall discrete polynomials was open during manydecades.

What we know for the differential case has seemed to be of little help becauseadding Dirac deltas to the classical discrete measures seems not to work.Indeed, Richard Askey in 1991 explicitly posed the problem of finding the firstexamples of Krall-discrete polynomials. He suggested to study measures whichconsist of some classical discrete weights together with a Dirac delta at the endpoint(s) of the interval of orthogonality. Three years later, Bavinck, vanHaeringen and Koekoek gave a negative answer to Askey’s question: theyadded a Dirac delta at zero to the Charlier and Meixner weights andconstructed difference operators with respect to which the correspondingorthogonal polynomials are eigenfunctions... but these difference operators havealways infinite order.In 2012, this speaker found the first examples of measures whose orthogonalpolynomials are also eigenfunctions of a higher order difference operator.




The problem of finding Krall discrete polynomials was open during manydecades.What we know for the differential case has seemed to be of little help becauseadding Dirac deltas to the classical discrete measures seems not to work.

Indeed, Richard Askey in 1991 explicitly posed the problem of finding the firstexamples of Krall-discrete polynomials. He suggested to study measures whichconsist of some classical discrete weights together with a Dirac delta at the endpoint(s) of the interval of orthogonality. Three years later, Bavinck, vanHaeringen and Koekoek gave a negative answer to Askey’s question: theyadded a Dirac delta at zero to the Charlier and Meixner weights andconstructed difference operators with respect to which the correspondingorthogonal polynomials are eigenfunctions... but these difference operators havealways infinite order.In 2012, this speaker found the first examples of measures whose orthogonalpolynomials are also eigenfunctions of a higher order difference operator.




The problem of finding Krall discrete polynomials was open during manydecades.What we know for the differential case has seemed to be of little help becauseadding Dirac deltas to the classical discrete measures seems not to work.Indeed, Richard Askey in 1991 explicitly posed the problem of finding the firstexamples of Krall-discrete polynomials. He suggested to study measures whichconsist of some classical discrete weights together with a Dirac delta at the endpoint(s) of the interval of orthogonality.

Three years later, Bavinck, vanHaeringen and Koekoek gave a negative answer to Askey’s question: theyadded a Dirac delta at zero to the Charlier and Meixner weights andconstructed difference operators with respect to which the correspondingorthogonal polynomials are eigenfunctions... but these difference operators havealways infinite order.In 2012, this speaker found the first examples of measures whose orthogonalpolynomials are also eigenfunctions of a higher order difference operator.




The problem of finding Krall discrete polynomials was open during manydecades.What we know for the differential case has seemed to be of little help becauseadding Dirac deltas to the classical discrete measures seems not to work.Indeed, Richard Askey in 1991 explicitly posed the problem of finding the firstexamples of Krall-discrete polynomials. He suggested to study measures whichconsist of some classical discrete weights together with a Dirac delta at the endpoint(s) of the interval of orthogonality. Three years later, Bavinck, vanHaeringen and Koekoek gave a negative answer to Askey’s question: theyadded a Dirac delta at zero to the Charlier and Meixner weights andconstructed difference operators with respect to which the correspondingorthogonal polynomials are eigenfunctions...

but these difference operators havealways infinite order.In 2012, this speaker found the first examples of measures whose orthogonalpolynomials are also eigenfunctions of a higher order difference operator.




The problem of finding Krall discrete polynomials was open during manydecades.What we know for the differential case has seemed to be of little help becauseadding Dirac deltas to the classical discrete measures seems not to work.Indeed, Richard Askey in 1991 explicitly posed the problem of finding the firstexamples of Krall-discrete polynomials. He suggested to study measures whichconsist of some classical discrete weights together with a Dirac delta at the endpoint(s) of the interval of orthogonality. Three years later, Bavinck, vanHaeringen and Koekoek gave a negative answer to Askey’s question: theyadded a Dirac delta at zero to the Charlier and Meixner weights andconstructed difference operators with respect to which the correspondingorthogonal polynomials are eigenfunctions... but these difference operators havealways infinite order.

In 2012, this speaker found the first examples of measures whose orthogonalpolynomials are also eigenfunctions of a higher order difference operator.




The problem of finding Krall discrete polynomials was open during manydecades.What we know for the differential case has seemed to be of little help becauseadding Dirac deltas to the classical discrete measures seems not to work.Indeed, Richard Askey in 1991 explicitly posed the problem of finding the firstexamples of Krall-discrete polynomials. He suggested to study measures whichconsist of some classical discrete weights together with a Dirac delta at the endpoint(s) of the interval of orthogonality. Three years later, Bavinck, vanHaeringen and Koekoek gave a negative answer to Askey’s question: theyadded a Dirac delta at zero to the Charlier and Meixner weights andconstructed difference operators with respect to which the correspondingorthogonal polynomials are eigenfunctions... but these difference operators havealways infinite order.In 2012, this speaker found the first examples of measures whose orthogonalpolynomials are also eigenfunctions of a higher order difference operator.


Krall discrete polynomials

How can we generate Krall discrete polynomials?

Apply a suitable Christoffel transform to the classical discrete measures, that is,multiply the classical discrete measures by suitable polynomials.

Depending on the classical discrete measure, we have found several classes ofsuitable polynomials for which this procedure works (1 class for Charlier, 2classes for Meixner and Krawtchouk and 4 classes for Hahn; the cross productof these classes also works).




Apply a suitable Christoffel transform to the classical discrete measures,

that is,multiply the classical discrete measures by suitable polynomials.





Apply a suitable Christoffel transform to the classical discrete measures, that is,multiply the classical discrete measures by suitable polynomials.



Krall-Charlier orthogonal polynomials

Consider the Charlier weight

µa =∞∑x=0

ax

x!δx ,

and a finite set F of positive integers. Orthogonal polynomials with respect to

µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx

are then eigenfunctions of a higher order difference operator. (Conjectured byA.J.D. in 2012, proved by myself for F = {1, 2, · · · , k} in 2013 and forarbitrary F with M.D. de la Iglesia in 2015).

The link with the symmetries for Casoratian determinants whose entries areCharlier polynomials is a consequence of the following fact:The orthogonal polynomials with respect to µF

a have two quite differentdeterminantal representations.




µa =∞∑x=0

ax

x!δx ,

and a finite set F of positive integers.

Orthogonal polynomials with respect to

µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx







µa =∞∑x=0

ax

x!δx ,


µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx

are then eigenfunctions of a higher order difference operator.

(Conjectured byA.J.D. in 2012, proved by myself for F = {1, 2, · · · , k} in 2013 and forarbitrary F with M.D. de la Iglesia in 2015).






µa =∞∑x=0

ax

x!δx ,


µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx

are then eigenfunctions of a higher order difference operator. (Conjectured byA.J.D. in 2012,

proved by myself for F = {1, 2, · · · , k} in 2013 and forarbitrary F with M.D. de la Iglesia in 2015).






µa =∞∑x=0

ax

x!δx ,


µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx

are then eigenfunctions of a higher order difference operator. (Conjectured byA.J.D. in 2012, proved by myself for F = {1, 2, · · · , k} in 2013

and forarbitrary F with M.D. de la Iglesia in 2015).






µa =∞∑x=0

ax

x!δx ,


µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx







µa =∞∑x=0

ax

x!δx ,


µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx


The link with the symmetries for Casoratian determinants whose entries areCharlier polynomials is a consequence of the following fact:

The orthogonal polynomials with respect to µFa have two quite different

determinantal representations.




µa =∞∑x=0

ax

x!δx ,


µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx





Christoffel transform

µ, (pn)nF = {f1, · · · , fk} a finite set of real numbers (increasing order).Christoffel transform µF of µ associated to F

µF = (x − f1) · · · (x − fk)µ.

Assume Φn = |pn+j−1(fi )|i,j=1,··· ,k 6= 0, then (Christoffel-Szego)

qn(x) =1

(x − f1) · · · (x − fk)

∣∣∣∣∣∣∣∣∣pn(x) pn+1(x) · · · pn+k(x)pn(f1) pn+1(f1) · · · pn+k(f1)

......

. . ....

pn(fk) pn+1(fk) · · · pn+k(fk)

∣∣∣∣∣∣∣∣∣are orthogonal with respect to µF .



µ, (pn)n

F = {f1, · · · , fk} a finite set of real numbers (increasing order).Christoffel transform µF of µ associated to F

µF = (x − f1) · · · (x − fk)µ.


qn(x) =1

(x − f1) · · · (x − fk)


......

. . ....





µ, (pn)nF = {f1, · · · , fk} a finite set of real numbers (increasing order).

Christoffel transform µF of µ associated to F

µF = (x − f1) · · · (x − fk)µ.


qn(x) =1

(x − f1) · · · (x − fk)


......

. . ....






µF = (x − f1) · · · (x − fk)µ.


qn(x) =1

(x − f1) · · · (x − fk)


......

. . ....






µF = (x − f1) · · · (x − fk)µ.

Assume Φn = |pn+j−1(fi )|i,j=1,··· ,k 6= 0,

then (Christoffel-Szego)

qn(x) =1

(x − f1) · · · (x − fk)


......

. . ....






µF = (x − f1) · · · (x − fk)µ.


qn(x) =1

(x − f1) · · · (x − fk)


......

. . ....





µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx

qa;Fn : orthogonal polynomials with respect to µF

a

qa;Fn (x) =

∣∣∣∣∣∣∣∣∣can(x) can+1(x) · · · can+k(x)can(f1) can+1(f1) · · · can+k(f1)

......

. . ....

can(fk) can+1(fk) · · · can+k(fk)

∣∣∣∣∣∣∣∣∣∏f∈F (x − f )

.

But there still is other determinantal representation for the orthogonalpolynomials with respect to µF

a .G = I (F ) = {g1, · · · , gm}, the polynomials

qa;Fn (x) =

∣∣∣∣∣∣∣∣∣can(x − fk − 1) −can−1(x − fk − 1) · · · (−1)mcan−m(x − fk − 1)c−ag1

(−n − 1) c−ag1

(−n) · · · c−ag1

(−n + m − 1)...

.... . .

...

c−agm (−n − 1) c−a

gm (−n) · · · c−agm (−n + m − 1)

∣∣∣∣∣∣∣∣∣ .are also orthogonal with respect to µF

a



µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx


a

qa;Fn (x) =


......

. . ....


∣∣∣∣∣∣∣∣∣∏f∈F (x − f )

.


a .

G = I (F ) = {g1, · · · , gm}, the polynomials

qa;Fn (x) =


(−n − 1) c−ag1

(−n) · · · c−ag1

(−n + m − 1)...

.... . .

...

c−agm (−n − 1) c−a

gm (−n) · · · c−agm (−n + m − 1)


a



µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx


a

qa;Fn (x) =


......

. . ....


∣∣∣∣∣∣∣∣∣∏f∈F (x − f )

.


a .One can get this representation by applying the method of D-operatorsdeveloped by myself and M.D. de la Iglesia. This method takes into accountthat the orthogonal polynomials with respect to µF

a are eigenfunctions of ahigher order difference operator.

G = I (F ) = {g1, · · · , gm}, the polynomials

qa;Fn (x) =


(−n − 1) c−ag1

(−n) · · · c−ag1

(−n + m − 1)...

.... . .

...

c−agm (−n − 1) c−a

gm (−n) · · · c−agm (−n + m − 1)


a



µFa =

∞∑x=0

∏f∈F

(x − f )ax

x!δx


a

qa;Fn (x) =


......

. . ....


∣∣∣∣∣∣∣∣∣∏f∈F (x − f )

.


a .G = I (F ) = {g1, · · · , gm}, the polynomials

qa;Fn (x) =


(−n − 1) c−ag1

(−n) · · · c−ag1

(−n + m − 1)...

.... . .

...

c−agm (−n − 1) c−a

gm (−n) · · · c−agm (−n + m − 1)


a



Since (qa;Fn )n and (qa;F

n )n are orthogonal polynomials with respect to the samemeasure µF

a and orthogonal polynomials with respect to a measure are uniqueup to multiplicative constants, we have that

qa;Fn (x) = γnq

a;Fn (x).

We can now explicitly compute the sequence γn by comparing

1 The leading coefficients of (qa;Fn )n and (qa;F

n )n.

2 The L2-norm of (qa;Fn )n and (qa;F

n )n.

In doing that, we get the invariance

CaF ,x = (−1)wF C

−aI (F ),−x

whereCaF ,x = det

(cafi (x + j − 1)

)ki,j=1


Meixner and Laguerre families

mn(x) ≡ Meixner polynomials (normalized mn(x) = xn/n! + · · · ).F = (F1,F2)

Ma,cF,x =

∣∣∣∣∣∣∣∣∣∣1≤j≤k[

ma,cf (x + j − 1)

]f ∈ F1[m

1/a,cf (x + j − 1)/aj−1

]f ∈ F2

∣∣∣∣∣∣∣∣∣∣a(k2

2 )−k2(k−1)(1− a)k1k2

.

I (F) = (I (F1), I (F2)), wF = wF1 + wF2

Ma,cF,x = (−1)wF M

a,−c−max F1−max F2I (F),−x

Lαn ≡ Laguerre polynomials

LαF,x = (−1)

∑F1

f

∣∣∣∣∣∣∣∣∣∣1≤j≤k[

(Lαf )(j−1)(x)]

f ∈ F1[Lα+j−1f (−x)

]f ∈ F2

∣∣∣∣∣∣∣∣∣∣.

LαF,x = (−1)wF L−α−max F1−max F2−2I (F),−x ,



mn(x) ≡ Meixner polynomials (normalized mn(x) = xn/n! + · · · ).

F = (F1,F2)

Ma,cF,x =

∣∣∣∣∣∣∣∣∣∣1≤j≤k[

ma,cf (x + j − 1)

]f ∈ F1[m

1/a,cf (x + j − 1)/aj−1

]f ∈ F2

∣∣∣∣∣∣∣∣∣∣a(k2

2 )−k2(k−1)(1− a)k1k2

.

I (F) = (I (F1), I (F2)), wF = wF1 + wF2




LαF,x = (−1)

∑F1

f

∣∣∣∣∣∣∣∣∣∣1≤j≤k[

(Lαf )(j−1)(x)]

f ∈ F1[Lα+j−1f (−x)

]f ∈ F2

∣∣∣∣∣∣∣∣∣∣.




mn(x) ≡ Meixner polynomials (normalized mn(x) = xn/n! + · · · ).F = (F1,F2)

Ma,cF,x =

∣∣∣∣∣∣∣∣∣∣1≤j≤k[

ma,cf (x + j − 1)

]f ∈ F1[m

1/a,cf (x + j − 1)/aj−1

]f ∈ F2

∣∣∣∣∣∣∣∣∣∣a(k2

2 )−k2(k−1)(1− a)k1k2

.

I (F) = (I (F1), I (F2)), wF = wF1 + wF2




LαF,x = (−1)

∑F1

f

∣∣∣∣∣∣∣∣∣∣1≤j≤k[

(Lαf )(j−1)(x)]

f ∈ F1[Lα+j−1f (−x)

]f ∈ F2

∣∣∣∣∣∣∣∣∣∣.



Exceptional polynomials

These symmetries are related to the equivalence between eigenstate adding anddeleting Darboux transformations for certain solvable quantum mechanicalsystems.

Exceptional polynomials (2007). They come from mathematical physics.These polynomials allow to write exact solutions of the Schrodinger equationcorresponding to certain rational extensions of the classical quantum potentials.Exceptional polynomials appeared some nine years ago, but there has been aremarkable activity around them mainly by theoretical physicists (withcontributions by D. Gomez-Ullate, N. Kamran and R. Milson, Y. Grandati, C.Quesne, S. Odake and R. Sasaki, A. Zhedanov, A.J.D. and many others).

pn, n ∈ X $ N, orthogonal and complete in L2(µ), that are also eigenfunctionsof a second order differential or difference operator (rational coefficients in theoperators)

Notice that Krall and Exceptional polynomials are coming from very differentproblems. For differential operators, no connection has been found betweenKrall and exceptional polynomials.Situation completely different at the discrete level: there is an unexpectedconnection between Krall and exceptional discrete polynomials, given by theduality between the variable and the index for the classical discrete families oforthogonal polynomials (A.J.D).



These symmetries are related to the equivalence between eigenstate adding anddeleting Darboux transformations for certain solvable quantum mechanicalsystems.Exceptional polynomials (2007).

They come from mathematical physics.These polynomials allow to write exact solutions of the Schrodinger equationcorresponding to certain rational extensions of the classical quantum potentials.Exceptional polynomials appeared some nine years ago, but there has been aremarkable activity around them mainly by theoretical physicists (withcontributions by D. Gomez-Ullate, N. Kamran and R. Milson, Y. Grandati, C.Quesne, S. Odake and R. Sasaki, A. Zhedanov, A.J.D. and many others).





These symmetries are related to the equivalence between eigenstate adding anddeleting Darboux transformations for certain solvable quantum mechanicalsystems.Exceptional polynomials (2007). They come from mathematical physics.

These polynomials allow to write exact solutions of the Schrodinger equationcorresponding to certain rational extensions of the classical quantum potentials.Exceptional polynomials appeared some nine years ago, but there has been aremarkable activity around them mainly by theoretical physicists (withcontributions by D. Gomez-Ullate, N. Kamran and R. Milson, Y. Grandati, C.Quesne, S. Odake and R. Sasaki, A. Zhedanov, A.J.D. and many others).





These symmetries are related to the equivalence between eigenstate adding anddeleting Darboux transformations for certain solvable quantum mechanicalsystems.Exceptional polynomials (2007). They come from mathematical physics.These polynomials allow to write exact solutions of the Schrodinger equationcorresponding to certain rational extensions of the classical quantum potentials.

Exceptional polynomials appeared some nine years ago, but there has been aremarkable activity around them mainly by theoretical physicists (withcontributions by D. Gomez-Ullate, N. Kamran and R. Milson, Y. Grandati, C.Quesne, S. Odake and R. Sasaki, A. Zhedanov, A.J.D. and many others).





These symmetries are related to the equivalence between eigenstate adding anddeleting Darboux transformations for certain solvable quantum mechanicalsystems.Exceptional polynomials (2007). They come from mathematical physics.These polynomials allow to write exact solutions of the Schrodinger equationcorresponding to certain rational extensions of the classical quantum potentials.Exceptional polynomials appeared some nine years ago, but there has been aremarkable activity around them mainly by theoretical physicists (withcontributions by D. Gomez-Ullate, N. Kamran and R. Milson, Y. Grandati, C.Quesne, S. Odake and R. Sasaki, A. Zhedanov, A.J.D. and many others).






pn,

n ∈ X $ N,

orthogonal

and complete in L2(µ),

that are also eigenfunctionsof a second order differential or difference operator

(rational coefficients in theoperators)





pn, n ∈ X $ N, orthogonal


that are also eigenfunctionsof a second order differential or difference operator

(rational coefficients in theoperators)





pn, n ∈ X $ N, orthogonal


that are also eigenfunctionsof a second order differential or difference operator (rational coefficients in theoperators)






Notice that Krall and Exceptional polynomials are coming from very differentproblems.

For differential operators, no connection has been found betweenKrall and exceptional polynomials.Situation completely different at the discrete level: there is an unexpectedconnection between Krall and exceptional discrete polynomials, given by theduality between the variable and the index for the classical discrete families oforthogonal polynomials (A.J.D).





Notice that Krall and Exceptional polynomials are coming from very differentproblems. For differential operators, no connection has been found betweenKrall and exceptional polynomials.

Situation completely different at the discrete level: there is an unexpectedconnection between Krall and exceptional discrete polynomials, given by theduality between the variable and the index for the classical discrete families oforthogonal polynomials (A.J.D).


Exceptional Hermite polynomials

Exceptional Hermite polynomials. Given a finite set of positive integersF = {f1, · · · , fk},

HFn (x) = Wr(Hn−uF ,Hf1 , · · · ,Hfk) =

∣∣∣∣∣∣∣∣∣∣Hn−uF (x) H ′n−uF (x) · · · H

(k)n−uF

(x)

Hf1 (x) H ′f1 (x) · · · H(k)f1

(x)...

.... . .

...

Hfk (x) H ′fk (x) · · · H(k)fk

(x)

∣∣∣∣∣∣∣∣∣∣uF =

∑f∈F

f −

(k + 1

2

)(deg(HF

n ) = n). n ∈ σF = {uF , uF + 1, · · · } \ (uF + F );

They allow to write exact solutions of the Schrodinger equation correspondingto the following rational extension of the harmonic oscillator potential (onedimensional).

VF = x2− d2

dx2(log(Wr(Hf1 , · · · ,Hfk ).





∣∣∣∣∣∣∣∣∣∣Hn−uF (x) H ′n−uF (x) · · · H

(k)n−uF

(x)

Hf1 (x) H ′f1 (x) · · · H(k)f1

(x)...

.... . .

...


(x)

∣∣∣∣∣∣∣∣∣∣

uF =∑f∈F

f −

(k + 1

2

)(deg(HF

n ) = n). n ∈ σF = {uF , uF + 1, · · · } \ (uF + F );


VF = x2− d2






∣∣∣∣∣∣∣∣∣∣Hn−uF (x) H ′n−uF (x) · · · H

(k)n−uF

(x)

Hf1 (x) H ′f1 (x) · · · H(k)f1

(x)...

.... . .

...


(x)

∣∣∣∣∣∣∣∣∣∣uF =

∑f∈F

f −

(k + 1

2

)(deg(HF

n ) = n).

n ∈ σF = {uF , uF + 1, · · · } \ (uF + F );


VF = x2− d2






∣∣∣∣∣∣∣∣∣∣Hn−uF (x) H ′n−uF (x) · · · H

(k)n−uF

(x)

Hf1 (x) H ′f1 (x) · · · H(k)f1

(x)...

.... . .

...


(x)

∣∣∣∣∣∣∣∣∣∣uF =

∑f∈F

f −

(k + 1

2

)(deg(HF

n ) = n). n ∈ σF = {uF , uF + 1, · · · } \ (uF + F );


VF = x2− d2




Exceptional Hermite polynomials HFn can be constructed from Charlier Krall

polynomials qa;Fn :

1 Dualizing qa;Fn (x)→ ca;F

n (x), where

ca;Fn (x), n ∈ {uF , uF + 1, · · · } \ (uF + F )

are exceptional Charlier polynomial: they are orthogonal andeigenfunctions of a second order difference operator (rational coefficients).

2 Passing to the limit when a goes to infinity ca;Fn (√

2ax + a)→ HFn (x).

HF ,x =1

2(k2)∏

f∈F f !det(H

(j−1)fi

(x))ki,j=1


This symmetry is related to the equivalence between eigenstate adding anddeleting Darboux transformation for the (onedimensional) harmonic oscillator.



Darboux transformation for the harmonic oscillator

H = − d2

dx2+ x2, V = x2

φ(x) = Hf (x)e−x2

H = BA B =d

dx− φ′

φA = − d

dx− φ′

φ

H = AB = − d2

dx2+ x2 − d2

dx2(log φ)

Exact solutions for this system are given by the Exceptional Hermitepolynomials H

{f }n (x)

F = {f1, · · · , fk}

V = x2 VF = x2 − d2

dx2log Wr(Hf1 , · · · ,Hfk )

Exact solutions for this system are given by the Exceptional Hermitepolynomials HF

n (x), n ∈ {uF , uF + 1, · · · } \ uF + F .Spectrum of energies EF = {uF , uF + 1, · · · } \ uF + F .

N \ ({0, 1, · · · , uF − 1} ∪ uF + F ) ≡ State deleting Darboux transformation.




H = − d2

dx2+ x2, V = x2


H = BA

B =d

dx− φ′

φA = − d

dx− φ′

φ

H = AB = − d2

dx2+ x2 − d2

dx2(log φ)


{f }n (x)

F = {f1, · · · , fk}

V = x2 VF = x2 − d2








H = − d2

dx2+ x2, V = x2


H = BA B =d

dx− φ′

φA = − d

dx− φ′

φ

H = AB = − d2

dx2+ x2 − d2

dx2(log φ)


{f }n (x)

F = {f1, · · · , fk}

V = x2 VF = x2 − d2








H = − d2

dx2+ x2, V = x2


H = BA B =d

dx− φ′

φA = − d

dx− φ′

φ

H = AB = − d2

dx2+ x2 − d2

dx2(log φ)


{f }n (x)

F = {f1, · · · , fk}

V = x2 VF = x2 − d2



n (x), n ∈ {uF , uF + 1, · · · } \ uF + F .

Spectrum of energies EF = {uF , uF + 1, · · · } \ uF + F .





H = − d2

dx2+ x2, V = x2


H = BA B =d

dx− φ′

φA = − d

dx− φ′

φ

H = AB = − d2

dx2+ x2 − d2

dx2(log φ)


{f }n (x)

F = {f1, · · · , fk}

V = x2 VF = x2 − d2







State deleting Darboux transformation

VF = x2 − d2

dx2(log Wr(Hf1 , · · · ,Hfk ) EF = {uF , uF + 1, · · · } \ uF + F .

State adding Darboux transformation.

H = − d2

dx2+ x2, V = x2

φ(x) = i fHf (−ix)ex2

≡ Virtual eigenfunction

G = {g1, · · · , gm}

V = x2 VG = x2 − d2

dx2log Wr(ig1Hg1 (−ix), · · · , igmHgm (−ix))

Spectrum of energies EG = {−g − 1, g ∈ G} ∪ N.

G = I (F ), EF and EG have the same structure

EF = uF + maxF + 1 + EG




VF = x2 − d2



H = − d2

dx2+ x2, V = x2



G = {g1, · · · , gm}

V = x2 VG = x2 − d2



G = I (F ),

EF and EG have the same structure





VF = x2 − d2



H = − d2

dx2+ x2, V = x2



G = {g1, · · · , gm}

V = x2 VG = x2 − d2



G = I (F ), EF and EG have the same structure



Selberg type integrals

By a Selberg type integrals we mean a multiple integral of the form∫· · ·∫

s(x)Λ2γ(x)dµ(x),

x stands for the multivariable x = (x1, · · · , xm), Λ(x) for the Vandermondedeterminant

Λ(x) =∏

1≤i<j≤m

(xi − xj),

dµ(x) = dµ(x1) · · · dµ(xm) and s(x) is a symmetric polynomial in the variablesx1, · · · , xm.When µ is the Jacobi measure dµ = xα−1(1− x)β−1dx , α, β > 0, ands(x) = 1, this integral is the celebrated Selberg integral (1944):∫ 1

0

· · ·∫ 1

0

Λ2γ(x)m∏j=1

xα−1j (1−xj)β−1dx =

m−1∏j=0

Γ(α + jγ)Γ(β + jγ)Γ(1 + (j + 1)γ)

Γ(α + β + (m + j − 1)γ)Γ(1 + γ).

Other choices of s(x) give well-known extensions of the Selberg integral. Forinstance, s(x) =

∏mj=1(xj − u) is Aomoto’s integral (1987), and

s(x) = Jλ(x1, · · · , xm, 1/γ), where γ ∈ N and Jλ is a Jack polynomial, isKadell’s integral (1988).

For s(x) = 1 and µ the Hermite measure dµ = e−x2/2dx , the integral is thealso celebrated Mehta integral (1967).




s(x)Λ2γ(x)dµ(x),


Λ(x) =∏

1≤i<j≤m

(xi − xj),

dµ(x) = dµ(x1) · · · dµ(xm) and s(x) is a symmetric polynomial in the variablesx1, · · · , xm.

When µ is the Jacobi measure dµ = xα−1(1− x)β−1dx , α, β > 0, ands(x) = 1, this integral is the celebrated Selberg integral (1944):∫ 1

0

· · ·∫ 1

0

Λ2γ(x)m∏j=1


m−1∏j=0

Γ(α + jγ)Γ(β + jγ)Γ(1 + (j + 1)γ)

Γ(α + β + (m + j − 1)γ)Γ(1 + γ).








s(x)Λ2γ(x)dµ(x),


Λ(x) =∏

1≤i<j≤m

(xi − xj),

dµ(x) = dµ(x1) · · · dµ(xm) and s(x) is a symmetric polynomial in the variablesx1, · · · , xm.When µ is the Jacobi measure dµ = xα−1(1− x)β−1dx , α, β > 0, ands(x) = 1, this integral is the celebrated Selberg integral (1944):∫ 1

0

· · ·∫ 1

0

Λ2γ(x)m∏j=1


m−1∏j=0

Γ(α + jγ)Γ(β + jγ)Γ(1 + (j + 1)γ)

Γ(α + β + (m + j − 1)γ)Γ(1 + γ).







F = {m,m + 1, · · · ,m + k}

Determinants with entries having a suitable integral representation can betransformed into multiple integrals.Heine (1878) If µn =

∫xndµ then

det((µi+j−2)ni,j=1) =

∫· · ·∫

Λ2(x)dµ(x).

Theorem (A.J.D., 2013) Fixed u ∈ R and consider an operator T , withdegree(T (p)) = degree(p)− 1. We associate to it the polynomials rn,u:deg(rn,u) = n, T (rn,u) = rn−1,u, rn,u(u) = 0, n ≥ 1. Assume that rn,u|rm,u ifn ≤ m.Let (pn)n be the sequence of monic orthogonal polynomials with respect to themeasure µ, then∫

· · ·∫

Λ2(x)m∏j=1

rn,u(xj)dµ(x) = CT ,n,m det(T i−1(pm+j−1(u))

)ni,j=1

,

where CT ,n,m = (−1)mnm!σmn

∏m−1j=0 ‖pj‖

2∏n−1j=0 σj .

The Theorem applies to T = d/dx ,∆ and many other important operators.



F = {m,m + 1, · · · ,m + k}Determinants with entries having a suitable integral representation can betransformed into multiple integrals.

Heine (1878) If µn =∫xndµ then


∫· · ·∫

Λ2(x)dµ(x).


· · ·∫

Λ2(x)m∏j=1


)ni,j=1

,


∏m−1j=0 ‖pj‖

2∏n−1j=0 σj .




F = {m,m + 1, · · · ,m + k}Determinants with entries having a suitable integral representation can betransformed into multiple integrals.Heine (1878) If µn =

∫xndµ then


∫· · ·∫

Λ2(x)dµ(x).


· · ·∫

Λ2(x)m∏j=1


)ni,j=1

,


∏m−1j=0 ‖pj‖

2∏n−1j=0 σj .





∫xndµ then


∫· · ·∫

Λ2(x)dµ(x).

Theorem (A.J.D., 2013) Fixed u ∈ R and consider an operator T , withdegree(T (p)) = degree(p)− 1. We associate to it the polynomials rn,u:deg(rn,u) = n, T (rn,u) = rn−1,u, rn,u(u) = 0, n ≥ 1. Assume that rn,u|rm,u ifn ≤ m.

Let (pn)n be the sequence of monic orthogonal polynomials with respect to themeasure µ, then∫

· · ·∫

Λ2(x)m∏j=1


)ni,j=1

,


∏m−1j=0 ‖pj‖

2∏n−1j=0 σj .





∫xndµ then


∫· · ·∫

Λ2(x)dµ(x).

Theorem (A.J.D., 2013) Fixed u ∈ R and consider an operator T , withdegree(T (p)) = degree(p)− 1. We associate to it the polynomials rn,u:deg(rn,u) = n, T (rn,u) = rn−1,u, rn,u(u) = 0, n ≥ 1. Assume that rn,u|rm,u ifn ≤ m.Let (pn)n be the sequence of monic orthogonal polynomials with respect to themeasure µ, then

∫· · ·∫

Λ2(x)m∏j=1


)ni,j=1

,


∏m−1j=0 ‖pj‖

2∏n−1j=0 σj .





∫xndµ then


∫· · ·∫

Λ2(x)dµ(x).


· · ·∫

Λ2(x)m∏j=1


)ni,j=1

,


∏m−1j=0 ‖pj‖

2∏n−1j=0 σj .



Constant term identities

Certain identities for constant term of Laurent polynomials are closely relatedto Selberg type integrals.The following one can be considered a contour version of Selberg integral. Itwas proposed by Morris (1982)

C.T.|z=0Λ2k(z)n∏

j=1

(1− zj

a

)xzm+k(n−1)j

= (−1)k(n2)+mna−mnn−1∏j=0

(x + kj)!(k(j + 1))!

(x −m + kj)!(m + kj)!k!,

wherez = (z1 · · · , zn)

and x ,m, k, n are nonnegative integers, m ≤ x and a 6= 0.Using some integral representations of classical and classical discreteorthogonal polynomials, we can transform some of our Wronskian typedeterminant into constant term identities.



Certain identities for constant term of Laurent polynomials are closely relatedto Selberg type integrals.

The following one can be considered a contour version of Selberg integral. Itwas proposed by Morris (1982)

C.T.|z=0Λ2k(z)n∏

j=1

(1− zj

a

)xzm+k(n−1)j

= (−1)k(n2)+mna−mnn−1∏j=0

(x + kj)!(k(j + 1))!

(x −m + kj)!(m + kj)!k!,

wherez = (z1 · · · , zn)





C.T.|z=0Λ2k(z)n∏

j=1

(1− zj

a

)xzm+k(n−1)j

= (−1)k(n2)+mna−mnn−1∏j=0

(x + kj)!(k(j + 1))!

(x −m + kj)!(m + kj)!k!,

wherez = (z1 · · · , zn)

and x ,m, k, n are nonnegative integers, m ≤ x and a 6= 0.

Using some integral representations of classical and classical discreteorthogonal polynomials, we can transform some of our Wronskian typedeterminant into constant term identities.




C.T.|z=0Λ2k(z)n∏

j=1

(1− zj

a

)xzm+k(n−1)j

= (−1)k(n2)+mna−mnn−1∏j=0

(x + kj)!(k(j + 1))!

(x −m + kj)!(m + kj)!k!,

wherez = (z1 · · · , zn)




Let us provide with an example. Consider the generating function for theMeixner polynomials(

1− z

a

)x(1− z)−x−c =

∞∑n=0

ma,cn (x)zn.

This gives

ma,cn (x) =

1

2πı

∫C

(1− z

a

)x(1− z)−x−c

zn+1dz ,

Using that integral representation for Meixner polynomials, we prove thefollowing constant term identity: (A.J.D. 2013)

C.T.|z=0

n∏j=1

(1− zj

a

)x(1− zj)x+c+n−1zm+n−1

j

Λ2(z)

= (−1)(m2)+(n2)+nmn!a(m2) det(ma,−c−n−i−j+3

n+i−1 (−x + j − 1))mi,j=1

.

For c = −x − n + 1, we recover Morris identity for k = 1. For m = 1, we get

C.T.|z=0

n∏j=1

(1− zj

a

)x(1− zj)x+c+n−1znj

Λ2(z) = (−1)(n+12 )n!ma,−c−n+1

n (−x).

which it can be considered a contour integral version of Aomoto’s integral.



Let us provide with an example.

Consider the generating function for theMeixner polynomials(

1− z

a

)x(1− z)−x−c =

∞∑n=0

ma,cn (x)zn.

This gives

ma,cn (x) =

1

2πı

∫C

(1− z

a

)x(1− z)−x−c

zn+1dz ,


C.T.|z=0

n∏j=1

(1− zj

a


j

Λ2(z)


n+i−1 (−x + j − 1))mi,j=1

.


C.T.|z=0

n∏j=1

(1− zj

a


Λ2(z) = (−1)(n+12 )n!ma,−c−n+1

n (−x).




Let us provide with an example. Consider the generating function for theMeixner polynomials(

1− z

a

)x(1− z)−x−c =

∞∑n=0

ma,cn (x)zn.

This gives

ma,cn (x) =

1

2πı

∫C

(1− z

a

)x(1− z)−x−c

zn+1dz ,


C.T.|z=0

n∏j=1

(1− zj

a


j

Λ2(z)


n+i−1 (−x + j − 1))mi,j=1

.


C.T.|z=0

n∏j=1

(1− zj

a


Λ2(z) = (−1)(n+12 )n!ma,−c−n+1

n (−x).



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