Jean-Marie Normand
email: jean-marie.normand@cea.fr

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. These kinds of determinants are encountered, for example, in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.

Introduction

This work has been motivated by studies of the probability density of the determinant (PDD) of random matrices [START_REF] Mehta | Probability density of the determinant of a random Hermitian matrix[END_REF][START_REF] Delannay | Distribution of the determinant of a random real-symmetric matrix from the Gaussian orthogonal ensemble[END_REF][START_REF] Normand | Probability density of the determinant of some random matrix ensembles preprint[END_REF]. The method used, and sketched in section 5, is to compute the Mellin transform of the PDD. In many cases it turned out to be a determinant with gamma functions as elements. One aim of this work is to evaluate some of these determinants and more generally determinants with shifted factorials (or Pochhammer's symbols) as elements.

We define in section 2 the s-shifted factorial (z) s;n , equation (2.1), as a generalization for non-negative values of n of the power function z n , the rising factorial (z) n , equation (2.3), and the falling factorial [z] n , equation (2.4); both the names and the notations of these last two objects are not well established, see [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF][START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF][START_REF] Comtet | Analyse combinatoire I and II[END_REF][START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF][START_REF] Roman | The Umbral Calculus[END_REF][START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF][START_REF] Rosen | Handbook of Discrete and Combinatorial Mathematics[END_REF] 1 . As a function of z, the s-shifted factorial is a special case of the polynomial sequences of the binomial type studied mainly in the calculus of finite differences and combinatorics; see in particular [START_REF] Comtet | Analyse combinatoire I and II[END_REF][START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF] and for a wide bibliography [START_REF] Roman | The Umbral Calculus[END_REF][START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF]. Expressed in terms of gamma functions, the s-shifted factorial can be extended to negative values and even complex values of n. The s-shifted factorial provides a compact formulation which emphasizes similarities and connections which exist between the power function and the shifted factorials: multiplication laws, Pascal triangle property, generating function and binomial formulae.

It is shown in section 3 that Vandermonde determinant with (z j ) s;i instead of (z j ) i as elements is still equal to the usual Vandermonde determinant. Other determinants with the inverse of a s-shifted factorial, or the ratio of two s-shifted factorials, as elements are also evaluated, both for positive and negative values of the index i. Using the relations between the s-shifted factorial and the gamma function, or the binomial coefficient, to each determinant evaluated in section 3, it corresponds a determinant in section 4 with elements expressed in terms of gamma functions. Finally, some applications of these determinants are given in section 5: evaluation of the PDD of random matrices and also, a possible application to Stieltjes moment problems 1 See [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] Pochhammer's symbol (z) n := z(z + 1) • • • (z + n -1) 6.1.22, [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF] (z) n := z(z + 1) • • • (z + n -1) p. xLiii, [START_REF] Comtet | Analyse combinatoire I and II[END_REF] 'factorielle z descendante d'ordre n' (z) n := z(z -1) • • • (z -n + 1) [4f], 'factorielle z montante d'ordre n' or Pochhammer's symbol n < z > n := z(z + 1) • • • (z + n -1) [4g], [START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF] lower factorial (z) n := z(z -1) • • • (z -n+1) (1.1), upper factorial z (n) := z(z +1) • • • (z +n-1) (1.2), [START_REF] Roman | The Umbral Calculus[END_REF] section 5 falling factorial sequence (z) n := z(z -1) • • • (z -n + 1) 2.1, rising factorial sequence < z > n := z(z + 1) • • • (z + n -1) 3.1, [START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF] falling factorial of length n [z] n := z(z -1) • • • (z -n+ 1) 3.2, rising factorial of length n [z] n := z(z + 1) • • • (z + n-1) 3.4 and III.2.A and [START_REF] Rosen | Handbook of Discrete and Combinatorial Mathematics[END_REF] z n := z(z -1) • • • (z -n + 1) nth falling power of z and z n := z(z + 1) • • • (z + n -1) nth rising power of z 3.4.2.

arising in connection with the boson normal ordering problem. As another example of application of the binomial formula, the finite sum of s-shifted factorials of arithmetic progression to n terms is evaluated in appendix A. Some basic properties of the product of differences and of the Vandermonde determinant are recalled, respectively, in appendices B and C. Finally, appendix D illustrates another way to handle s-shifted factorials.

2. Definition and some properties of the s-shifted factorial 2.1. Definitions and relations between shifted factorials With n a non-negative integer, z and s (the shift) some complex numbers, let us define the s-shifted factorial by (z) s;n := 1 n = 0

z(z + s) • • • z + (n -1)s n = 1, 2, . . . . ( 2.1)
For s = 0, 1 and -1, this definition coincides, respectively, with the power function, the rising factorial (or Pochhammer's symbol, mainly in hypergeometric theory) and the falling factorial, namely for n nonzero, (z) 0;n = z n (2.2)

(z) 1;n = (z) n := z(z + 1) • • • (z + n -1) (2.3) (z) -1;n = [z] n := z(z -1) • • • (z -n + 1)
(2.4)

and when n = 0 all these quantities take the value 1. Thereby, the s-shifted factorial allows compact expressions which emphasize the similarities between the power function and the shifted factorials.

For any non-negative integer n, one has

(z) s;n = (-1) n (-z) -s;n (2.5) = z + (n -1)s -s;n . (2.6)
For s nonzero, the s-shifted factorials are related to the rising factorial, equation (2.3), by

(z) s;n = s n z s n . (2.7)
As a function of z, (z) s;n is a monic polynomial (i.e. the coefficient of the highest power is one) in z of degree n,

(z) s;n = z n + n(n -1) 2 s z n-1 + • • • + (n -1)! s n-1 z (2.8)
with 0, -s, . . . , -(n -1) s as zeros. Consequences of these properties in terms of Vandermonde determinants are developed in section 3. The sets of polynomials {(z) s;n , n = 0, 1, . . .} are special cases of the polynomial sequences {p n (z), n = 0, 1, . . .}, p n (z) being exactly of degree n. We are going to use these sequences in the way it is done in combinatorics [7-9] 2 . Any polynomial sequence is a basis of the vector space P over the complex field of complex polynomials in the variable z. Then, to any two polynomial sequences {p n (z)} and {q n (z)} there exist uniquely determined connecting coefficients such that q n (z) = n k=0 c n,k p k (z). These important coefficients have been widely studied, e.g.,

[z] n = n k=0 s(n, k) z k z n = n k=0 S(n, k) [z] k (z) n = n k=0 L(n, k) [z] k (2.9)
where s(n, k), S(n, k) and L(n, k) = n-1 k-1 n!/k! are, respectively, the Stirling numbers of the first and second kind [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF][START_REF] Comtet | Analyse combinatoire I and II[END_REF][START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF][START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF][START_REF] Rosen | Handbook of Discrete and Combinatorial Mathematics[END_REF] 3 and the signless Lah numbers [START_REF] Comtet | Analyse combinatoire I and II[END_REF][START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF][START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF][START_REF] Rosen | Handbook of Discrete and Combinatorial Mathematics[END_REF] 4 (other relations between z n , (z) n and [z] n 2 See, e.g., [START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF] section 1, [8] section 3, [START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF] section III.2. 3 See, e.g., [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 24.1.3,4, [START_REF] Comtet | Analyse combinatoire I and II[END_REF] chapter V [5e,f], [START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF] (1.11-13), [START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF] 3.24,25 or [10] 2.5.2. 4 See, e.g., [START_REF] Comtet | Analyse combinatoire I and II[END_REF] chapter III p. 165 , [START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF] (1.14) and section 9, [START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF] 3.24,25, [10] 3.1.8. immediately follow from equations (2.5) and (2.6)). We will see in subsection 2.6 that {(z) s;n , n = 0, 1, . . .} has in addition the important property to be a polynomial sequence of the binomial type.

Special values

With k some non-negative integer, one gets

(-k) n = (-1) n [k] n =    0 k = 0, . . . , n -1 (-1) n k! (k -n)! k = n, n + 1, . . . (2.10) (k) n = (-1) n [-k] n =    0 k = 0 (k + n -1)! (k -1)! k = 1, 2, . . . . ( 2
(z) n = Γ(z + n) Γ(z) = (z + n -1)! (z -1)! = (-1) n n! -z n (2.12) [z] n = Γ(z + 1) Γ(z -n + 1) = z! (z -n)! = n! z n . (2.13)
For s nonzero, relations with (z) s;n follow from equation (2.7).

Actually, the relations above can be taken as the definition of (z) s;n in terms of the gamma function. Thereby, one extends the s-shifted factorial to negative values, and even to complex values t of n, defining the generalized s-shifted factorial by

(z) s;t := s t Γ z s + t Γ z s (2.14)
with a cut, say, along the negative real axis of the complex s plane, with -π < arg s ≤ π, to ensure a single-valued dependence on s, and we choose the determination such that s t = 1 for s = 1. As s goes to zero, say, along the real axis, using the Stirling formula [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 6 , Γ(z) ∼ e -z z z-1 2 (2π) 5 See, e.g., [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 6.1.5, 6.1.21 and 6.1.22. 6 See, e.g., [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 6.1.37. 7 See, e.g., [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 6.1.17.

Thus for any integer q, (z) s;q = (-1) q (-z) -s;q (2.20) = z + (q -1)s -s;q (2.21) which generalize equations (2.5) and (2.6) for any integer, even negative.

Multiplication laws

When the power function fulfils z t z r = z t+r with r and t some complex numbers, it follows from the definition (2. This relation generalizes z t = 1/z -t for s = 0 and provides the relation between the s-shifted factorials for any integer q and -q, by equation (2.21),

(z) s;-q = 1 (z -qs) s;q = 1 (z -qs) z -(q -1)s • • • (z -s) = 1 (z -s) -s;q . (2.24)
In terms of binomial coefficients, the multiplication law (2.22) reads

z n z -n p = n + p n z n + p or z n [n] p = [z] p z -p n -p . (2.25)
For a proper choice of determination the power function fulfils w t z t = (wz) t . For the s-shifted factorial, one has (wz) s;t = s t ( s w ) t (z) s w ;t (2.26) and thus for any integer q, (wz) s;q = w q (z) s w ;q .

(2.27)

For w = -1, relation (2.26) corresponds to equation (2.17). For w = k and q = n some non-negative integers, iterating the multiplication law (2.22) and from equation (2.27), (kz

) s;kn = k kn k-1 ℓ=0 n-1 j=0 z + (nℓ + j) s k = k kn k-1 ℓ=0 z + ℓ k s s;n (2.28)
where the last equality corresponds to a rearrangement of the factors, both nℓ + j and ℓ + jk taking once all the kn values 0, 1, . . . , kn -1. The equation above can also be obtained from the definition (2.14) and the Gauss multiplication formula [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 8 , Γ(kz) = (2π) Now, the generating function of the rising factorials can be obtained directly from the binomial series with equation (2.12),

1 2 (1-k) k kz-1 2 k-1 ℓ=0 Γ z + ℓ/k . Note
(1 -x) -z = ∞ n=0 (-1) n -z n x n = ∞ n=0 (z) n x n n! = G 1;z (x) , |x| < 1 . (2.38) Therefore, G s;z (x) = (1 -sx) -z s (2.39) recovering for s = 0 the expression G 0,z (x) := ∞ n=0 z n x n n! = e xz .
Since the generating function G s;z (x) above reads as an exponential function F(x) z of z, it satisfies the multiplication law [START_REF] Moussa | [END_REF] G s;z (x) G s;w (x) = G s;z+w (x) .

(2.40)

Expanding both sides of this last equation as a power series in x yields,

(z + w) s;n = n k=0 n k (z) s;k (w) s;n-k (2.41)
namely, the s-shifted factorial satisfies the binomial formula. The polynomial sequence {(z) s;n , n = 0, 1, . . .} which satisfies (z) s;0 = 1 and the binomial formula above is said to be of binomial type [START_REF] Comtet | Analyse combinatoire I and II[END_REF][START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF][START_REF] Roman | The Umbral Calculus[END_REF][START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF][START_REF] Rosen | Handbook of Discrete and Combinatorial Mathematics[END_REF] 10 . This property is shared by many other binomial sequences {p n (z), n = 0, 1, . . .} which have been studied mainly in combinatorics using generating function methods and above all efficient operator methods.

The binomial sequences can be characterized by a generating function which depends exponentially on

z [9] 11 G z (x) = e g(x)z = e (x+g2x 2 +•••) z = ∞ n=0 p {g} n (z)
x n n! (2.42) then p {g} n (z) is a monic polynomial of degree n in z, the coefficients of which are known as Bell polynomials [6] 12 (indeed, expanding the exponential series above, the term in z n reads z n x n 1 + O(x) /n!). In the case we consider, g(x) := -s -1 ln(1 -sx) and p {g} n (z) = (z) s;n ; for s = 0, g(x) = x and p {g} n (z) = z n . The binomial sequences can as well be characterized by the fact [START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF][START_REF] Roman | The Umbral Calculus[END_REF][START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF] 13 that the basis operator of the sequence, i.e. the linear operator D of the vector space P (already considered in subsection 2.1) into itself defined by Dp 0 (z) := 0 and Dp n (z) := np n-1 (z) for n ≥ 1, is a delta operator, i.e. it is shift invariant, DE a = E a D for all complex number a, where E a is the translation operator defined by E a f(z) := f(a + z) and moreover Dz = c = 0. In our case, from equation (2.34), D s := -∆ -s /s = (I-E -s )/s, where I is the identity operator, is clearly shift invariant and D s z = 1. Indeed, the binomial formula (2.41) for the s-shifted factorial can also be demonstrated by recurrence from the Pascal triangle properties (2.31) and (2.32). It is true for n = 0 and 1. Let us assume it to be true for n, then,

(z + w) s;n+1 = n k=0 n k (z) s;k (w) s;n-k z + ks + w + (n -k)s = n+1 k=0 n k -1 (z) s;k (w) s;n+1-k + n+1 k=0 n k (z) s;k (w) s;n+1-k = n+1 k=0 n + 1 k (z) s;k (w) s;n+1-k . (2.43)
By equation (2.12), in terms of binomial coefficients, the binomial formula (2.41) reads

z + w n = n k=0 z k w n -k . ( 2

.44)

As for the power function, the binomial formula for the s-shifted factorial can be directly extended to p > 2 variables using multinomial coefficients,

p j=1 z j s;n = n n 1 ,...,np=0 n 1 +•••+np=n n! n 1 ! • • • n p ! (z 1 ) s;n1 • • • (z p ) s;np .
(2.45) 10 See, e.g., [START_REF] Comtet | Analyse combinatoire I and II[END_REF] [6a] and [13c,d], [START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF] (1.6), [START_REF] Roman | The Umbral Calculus[END_REF] section 5, [START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF] section III.2, [10] 3.4.2 (3.). 11 See, e.g., [START_REF] Aigner | Combinatorial Theory Reprint of the[END_REF] 3.59. 12 See, e.g., [START_REF] Comtet | Analyse combinatoire I and II[END_REF] section III.3. 13 See, e.g., [START_REF] Rota | On the Foundations of Combinatorial Theory: III. Theory of Binomial Enumeration[END_REF] section 3 theorem 1, [8] section 7, [9] 3.45.

From equation (2.5) the following corollary is immediately obtained:

(z -w) s;n = n k=0 (-1) n-k n k (z) s;k (w) -s;n-k . (2.46)
Although, as already noted, (z -1 ) i = ((z) i ) -1 , the binomial formula can be extended to the inverse of s-shifted factorials. Indeed, by equations (2.30), (2.41) and (2.6),

n k=0 n k 1 (z) s;k 1 (w) s;n-k = n k=0 n k z + (n -1)s -s;n-k (z) s;n w + (n -1)s -s;k (w) s;n = z + w + 2(n -1)s -s;n (z) s;n (w) s;n = z + w + (n -1)s s;n (z) s;n (w) s;n (2.47) corresponding for s = 0 to n k=0 n k 1 z k 1 w n-k = (z + w) n z n w n = 1 z + 1 w n (2.48)
where, once again, the last equality above does not hold for s nonzero. Similarly, one also gets from equations (2.30) and (2.46)

n k=0 (-1) n-k n k (z) s;k (w) s;k = n k=0 (-1) n-k n k (z) s;k w + (n -1)s -s;n-k (w) s;n = z -w -(n -1)s s;n (w) s;n (2.49) corresponding for s = 0 to n k=0 (-1) n-k n k z k w k = (z -w) n w n = z w -1 n .
(2.50)

Another kind of useful relations is as follows. From equation (2.25), the multiplication law (2.22) and the binomial formula (2.41),

n k=0 n k [k] p (z) s;k (w) s;n-k = [n] p n k=p n -p k -p (z) s;k (w) s;n-k = [n] p (z) s;p n k=p n -p k -p (z + ps) s;k-p (w) s;n-k = [n] p (z) s;p (z + w + ps) s;n-p . (2.51)
Several of these binomial formulae are used in the next section to evaluate some determinants with s-shifted factorials as elements. As another example of application, the finite sum of s-shifted factorials of arithmetic progression to n terms is evaluated in appendix A.

Generalized Vandermonde determinant with s-shifted factorials as elements

In what follows, n is a positive integer and z is, either a set of complex numbers, or a complex function, z := {z j , j = 0, . . . , n -1} or j → z(j) := z j j = 0, . . . , n -1 .

(3.1) Some basic properties of the product of differences ∆ n (z) := 0≤i<j≤n-1 (z j -z i ), equation (B.1), and of the Vandermonde determinant det (z j ) i i,j=0,...,n-1 , are recalled, respectively, in appendices B and C.

3.1. Expressions for s-shifted factorial with a non-negative index Lemma 1. With n a positive integer and s some complex number, the generalized Vandermonde determinant of s-shifted factorials, still is the product of differences,

det (z j ) s;i i,j=0,...,n-1 = ∆ n (z) (3.2)
thus, it does not depends on s. More generally,

det Π i (z j ) i,j=0,...,n-1 = λ ∆ n (z) (3.3)
where Π i (z) are n linearly independent polynomials in (z) s;. each of degree less than n and defined as follows:

Π i (z) := n-1 k=0 c i,k (z) s;k i = 0, . . . , n -1 λ := det c i,k i,k=0,...,n-1 = 0 . (3.4)
In particular, with b i some complex numbers, one has

det (b i + z j ) s;i i,j=0,...,n-1 = ∆ n (z) . (3.5)
Finally, with t some complex number,

det (z j ) s;t+i i,j=0,...,n-1 = n-1 j=0 (z j ) s;t ∆ n (z) . (3.6)
Two proofs are given. Based on the properties of the s-shifted factorial, proof 1 expresses the determinants considered in terms of Vandermonde determinants. Illustrating again the similarities between (z) s;i and z i , proof 2 follows the same steps as a usual way of computing the Vandermonde determinant, equation (C.1).

Proof 1. The s-shifted factorial (z) s;i is a monic polynomial of degree i in z, see equation (2.8). Hence, formula (3.2) follows from equation (C.3). Note that equation (3.2) still holds for the monic polynomials obtained from any generating function defined by equation (2.42). Formulae (3.3) and (3.5) can be directly obtained either from equation (C.3) in terms of usual polynomials (e.g., (b i + z) s;i is also a monic polynomial of degree i in z) or starting from formula (3.2), by the same arguments as for equation (C.3), in terms of polynomials in s-shifted factorials (e.g., by the binomial formula (2.41), (b i + z) s;i = i k=0 i k (b i ) s;k (z) s;n-k , i.e. a monic polynomial of degree i in (z) s;. ). Finally, equation (3.6) follows from the multiplication law (2.22) and formula (3.5).

Proof 2. Let M i,j := (z j ) s;i . The determinant det[M i,j ] i,j=0,...,n-1 is not changed if one replaces the row R i by the linear combination R i -(M i,0 /M i-1,0 ) R i-1 , successively for i = n -1, n -2, . . . , 1.
Then, by the multiplication law (2.22), for i = 1, . . . , n -1 and j = 0, . . . , n -1,

M i,j → (z j ) s;i -z 0 + (i -1)s (z j ) s;i-1 = (z j -z 0 ) (z j ) s;i-1 . (3.7)
This operation replaces the column C 0 by zeros except for the row R 0 left unchanged. Expanding the determinant with respect to C 0 and taking out the factors depending only on j yield the recurrence formula on n, D s;n (z 0 , . . . , z n-1 ) := det M i,j i,j=0,...,n-1

= n-1 j=1 (z j -z 0 ) D s;n-1 (z 1 , . . . , z n-1 ) . (3.8)
An iteration of this last equation, down to D s;1 (z n-1 ) = 1, completes the proof. The recurrence procedure above makes step by step the matrix (M i,j ) i,j=0,...,n-1 triangular. Let us denote by a superscript the rank of the step in this procedure. At the first step the row i = 0 is unchanged while for i = 1, . . . , n -1,

R (1) i = R i - M i,0 M i-1,0 R i-1 . (3.9)
At the second step the rows i = 0, 1 are unchanged, while for i = 2, . . . , n -1, R

i = R (2)
i -

M (1) i,1 M (1) i-1,1 R (1) i-1 = R i - M i,0 M i-1,0 + M (1) i,1 M (1) i-1,1 R i-1 + M (1) i,1 M (1) i-1,1 M i-1,0 M i-2,0 R i-2 . (3.10)
Generically, the final expression of the row i is given by R (i)

i . It happens that in the special case z j := b + js, with b some complex number and s nonzero, these expressions read

R (1) i = R i -b + (i -1)s R i-1 (3.11) R (2) i = R i -2 b + (i -1)s R i-1 + b + (i -1)s b + (i -2)s R i-2 (3.12) . . . R (i) i = i k=0 (-1) i-k i k b + (i -1)s -s;i-k R k (3.13)
where the last formula can be checked as follows. With M i,j := (b + js) s;i , by the binomial formula (2.46), equations (2.7) and (2.6),

M (i) i,j = i k=0 (-1) i-k i k b + (i -1)s -s;i-k (b + js) s;k = s i (j -i + 1) i = s i [j] i (3.14)
which vanishes for i > j, see equation (2.10). Another proof of this identity is given in appendix D.1. Thus, as expected, the resulting matrix is triangular and its determinant is the product of its diagonal elements s j [j] j = s j j!. Then, by equation (B.5),

det (b + js) s;i i,j=0,...,n-1 = s n(n-1)/2 n-1 j=0 j! = ∆ n (j → b + js) (3.15)
completing the proof of equation (3.2) in the special case z j := b + js.

Lemma 2. With n a positive integer, s some complex number and z j = 0, -s, . . . ,

-(n -2)s, det 1 (z j ) s;i i,j=0,...,n-1 = (-1) n(n-1)/2 n-1 j=0 (z j ) s;n-1 ∆ n (z) . (3.16)
This formula generalizes equation (C.5).

Proof 1. With n -1 ≥ i ≥ 0 and z j = 0, -s, . . . , -(n -2)s, by equations (2.30) one gets det 1 (z j ) s;i i,j=0,...,n-1

= det z j + (n -2)s -s;n-1-i i,j=0,...,n-1 n-1 j=0 (z j ) s;n-1 . (3.17)
Then, changing i into n -1 -i (i.e. rearranging the rows) on the right-hand side determinant above, lemma 2 follows from equation (3.5). Note that when s = 0, equation (C.5) for the power function can also be derived as above from 1/z i = z n-1-i /z n-1 .

Proof 2. This proof of lemma 2 follows the same steps as proof 2 of lemma 1. With the linear combination of rows R i -

(M i,0 /M i-1,0 ) R i-1 , M i,j := 1 (z j ) s;i → -(z j -z 0 ) z 0 + (i -1)s z j 1 (z j + s) s;i-1 , i = 1, . . . , n -1 . (3.18)
Then, with z j = 0, -s, . . . , -(n -2)s, the recurrence formula on n reads

D s;n (z 0 , . . . , z n-1 ) := det M i,j i,j=0,...,n-1 = (-1) n-1 n-1 j=1 (z j -z 0 ) (z 0 ) s;n-1 n-1 j=1 z j D s;n-1 (z 1 + s, . . . , z n-1 + s) . (3.19)
Iteration of this last equation, down to D s;1 z n-1 + (n -1)s = 1, ends the proof.

As in proof 2 of lemma 1, in the special case z j := b + js with b some complex number and s nonzero, the determinant can be made triangular in one step, replacing R i by the linear combination

R (i) i = i k=0 i k 1 -b -2(i -1)s s;i-k R k . (3.20)
Indeed, with M i,j := 1/(b + js) s;i and b = 0, -s, . . . , -(2n -3)s, it follows from the binomial formula (2.47) and equations (2.5)-(2.7)

M (i) i,j = i k=0 i k 1 -b -2(i -1)s s;i-k 1 (b + js) s;k = (-s) i [j] i b + (i -1)s s;i (b + js) s;i (3.21)
which vanishes for i > j. Another proof of this identity is given in appendix D.2. Then, the determinant is the product of its diagonal elements,

;i (z + b + is) s;n-1-i = n-1-i k=0 n -1 -i k (b) s;k (z) s;i (z + is) s;n-1-i-k = n-1-i k=0 n -1 -i k (b) s;k (z) s;n-1-k = Π i (z) (3.27)
where Π i (z), a polynomial in (z) s;. of degree n -1, is defined as in equation (3.4) with

c i,k :=    0 k = 0, . . . , i -1 n -1 -i n -1 -k (b) s;n-1-k k = i, . . . , n -1 .
(3.28)

Thus, the matrix [c i,k ] i,k=0,...,n-1 is triangular and its determinant is the product of its diagonal elements,

det[c i,k ] i,k=0,...,n-1 = n-1 j=0 (b) s;j . (3.29)
Then, when a = 1, lemma 3 follows from equation (3.3), with

z j + b = 0, -s, . . . , -(n -2)s, det (z j ) s;i (z j + b) s;i i,j=0,...,n-1 = n-1 j=0 1 (z j + b) s;n-1 det[Π i (z j )] i,j=0,...,n-1 = n-1 j=0 (b) s;j (z j + b) s;n-1
∆ n (z) .

(3.30) Note that for s = 0, the equation (C.6) for the power function can also be derived as above. When a = 1, (z) s;i (az + b + is) s;n-1-i is still a polynomial Π i (z) of degree n -1 in (z) s;. (or equivalently z).

But now the evaluation of the connecting coefficients c i,k (or even to compute det[c i,k ] i,k=0,...,n-1 we only need) is no longer easy since there is no simple combination law between the s-shifted factorials of z and az. proof 2 provides a simple proof of lemma 3 for all values of a.

Proof 2. This proof of equation (3.25) follows the same steps as proof 2 of lemma 1. With the linear combinations of rows R i -

(M i,0 /M i-1,0 ) R i-1 , M i,j := (z j ) s;i (az j + b) s;i → (z j -z 0 ) b + (i -1)(1 -a)s az 0 + b + (i -1)s (az j + b) (z j ) s;i-1 (az j + b + s) s;i-1 , i = 1, . . . , n -1 . (3.31)
Then, with az j + b = 0, -s, . . . , -(n -2)s, the recurrence formula on n reads Note that in lemma 3, corresponding to a rearrangement of the factors,

n-1 j=0 b + (n -1 -j)(1 -a)s s;j = n-1 j=0 b + (n -1 -j)s (1-a)s;j . (3.33)
As in proof 2 of lemma 1, in the special case z j := c + js with c some complex number, a = 1 and s nonzero, the determinant can be made triangular in one step, replacing R i by the linear combination

R (i) i = i k=0 (-1) i-k i k c + (i -1)s -s;i-k d + 2(i -1)s -s;i-k R k (3.34)
where d := b + c. Indeed, with M i,j := (c + js) s;i /(d + js) s;i and d = 0, -s, . . . , -(2n -3)s, after some elementary algebra based on the relations (2.5)-(2.7), (2.12) and (2.13), one gets

M (i) i,j = i k=0 (-1) i-k i k c + (i -1)s -s;i-k d + 2(i -1)s -s;i-k (c + js) s;k (d + js) s;k = (-1) i (c) s;i d + (i -1)s s;i 3 F 2 (cs -1 + j, ds -1 + i -1, -i; cs -1 , ds -1 + j; 1) (3.35)
where the 3 F 2 is a terminating Saalschützian generalized hypergeometric series [START_REF] Bateman | Higher Transcendental Functions[END_REF] 14 ,

3 F 2 (α, β, -i; γ, 1 + α + β -γ -i; 1) = (γ -α) i (γ -β) i (γ) i (γ -α -β) i i = 0, 1, . . . . (3.36)
Hence,

M (i) i,j = s i [j] i (d -c
s j j! (d -c) s;j (d + js) s;n-1 = n-1 j=0 (d -c) s;j (d + js) s;n-1 ∆ n (j → c + js) (3.38)
ending the proof of equation (3.25) in the special case z j := c + js and a = 1.

Consequences for s-shifted factorial with a negative index

Using equation (2.24), (z) s;-i = 1/(z -s) -s;i , and (B.2), the following corollaries are direct consequences of the previous lemmas.

Corollary 1.

With n a positive integer, s some complex number and z j = s, 2s, . . . , (n -1)s, det (z j ) s;-i i,j=0,...,n-1 = det 1 (z j -s) -s;i i,j=0,...,n-1 = (-1) n(n-1)/2 n-1 j=0 (z j ) s;-(n-1) ∆ n (z) . (3.39)

Proof. Consequence of lemma 2. 14 See, e.g., [START_REF] Bateman | Higher Transcendental Functions[END_REF] equations 2.1(30) and 4.4(3).

Corollary 2.

With n a positive integer and s some complex number, det 1 (z j ) s;-i i,j=0,...,n-1 = det (z j ) s;i i,j=0,...,n-1 = ∆ n (z) .

(3.40)

Proof. Consequence of lemma 1.

Corollary 3.

With n a positive integer, a, b and s some complex numbers and az j + b = s, 2s, . . . , (n -1)s, (ii) A proof following the same steps as proof 2 of lemma 1, and using the same linear combination of rows and also, with t some complex number, det (z j ) s;t-i i,j=0,...,n-1 = n-1 j=0 (z j ) s;t det (z j + t) s;-i i,j=0,...,n-1 .

det (az j + b) s;-i (z j ) s;-i i,j=0,...,n-1 = det (z j -s) -s;i (az j + b -s) -s;i i,j=0,...,n-1 = n-1 j=0 (az j + b) s;-(n-1) b + s + (n -j)(a -1)s s;-j ∆ n (z) . ( 3
R i -(M i,0 /M i-1,0 ) R i-1 ,
(3.44) Lemma 4. With n a positive integer and s some complex number,

det (z i + w j ) s;n-1 i,j=0,...,n-1 = (-1) n(n-1)/2 (n -1)! n n-1 j=0 j! 2 ∆ n (z) ∆ n (w) . ( 3

.45)

Proof. By the binomial formula (2.41), with M i,j := (z i + w j ) s;n-1 = n-1 k=0

n-1 k (z i ) s;k (w j ) s;n-1-k , the matrix M reads as the product of two matrices. Since the determinant of the product is the product of the determinants, one gets det (z i + w j ) s;n-1 i,j=0,...,n-1 = det n -1 k (z i ) s;k i,k=0,...,n-1 det (w j ) s;n-1-k j,k=0,...,n-1 .

(3.46)

Taking the binomial coefficients out of the first determinant and rearranging the rows of the last determinant, equation (3.45) follows from lemma 1.

Note that in all lemmas and corollaries above, the determinants considered are anti-symmetric polynomials or rational fractions of the n variables z 0 , . . . , z n-1 , therefore one expects the simplest polynomial alternant ∆ n (z) to be a factor of the result. The same argument holds for ∆ n (w) in lemma 4.

Determinants with gamma functions or binomial coefficients as elements

Using the relations (2.12)-(2.14) between the s-shifted factorial and the gamma function or the binomial coefficient, the results listed below are immediate consequences of the formulae derived in section 3 with s = ±1. For corollaries 4-6, a direct proof following the same steps as proof of lemma 1, and using the same linear combination of rows R i -(M i,0 /M i-1,0 ) R i-1 , can also be given. It is only sketched as an example for corollary 4. In the special case z j = b + aj, with a and b some complex numbers, the product of differences ∆ n (z) is given by equation (B.5).

Corollary 4.

With n a positive integer,

det Γ(z j + i) i,j=0,...,n-1 = n-1 j=0 Γ(z j ) ∆ n (z) z j = 0, -1, . . . (4.1) det z j i i,j=0,...,n-1 = 1 n-1 j=0 j! ∆ n (z) . (4.2) Proof 1. Consequences of lemma 1. Proof 2. With the linear combination of rows R i -(M i,0 /M i-1,0 ) R i-1 , M i,j := Γ(z j + i) → (z j -z 0 ) Γ(z j + i -1) i = 1, . . . , n -1 . (4.3)
Then, with z j = 0, -1, . . ., the recurrence formula on n reads D n (z 0 , . . . , z n-1 ) := det M i,j i,j=0,...,n-1

= Γ(z 0 ) n-1 j=1 (z j -z 0 ) D n-1 (z 1 , . . . , z n-1 ) . ( 4
det 1 Γ(z j + i) i,j=0,...,n-1 = (-1) n(n-1)/2 n-1 j=0 Γ(z j + n -1) ∆ n (z) (4.6)
and for z j = 0, 1, . . . , n -2, det 1

zj i i,j=0,...,n-1 = (-1) n(n-1)/2 n-1 j=0 j! [z j ] n-1 ∆ n (z) . (4.7)
Proof. Consequences of lemma 2.

In the special case z j := b + j, one gets [START_REF] Mehta | [END_REF]

det 1 Γ(b + i + j) i,j=0,...,n-1 = (-1) n(n-1)/2 n-1 j=0 j! Γ(b + n -1 + j) . (4.8)
Corollary 6. With n a positive integer and b some complex numbers, for z j = 0, -1, . . .,

det Γ(z j + i) Γ(az j + b + i) i,j=0,...,n-1 = n-1 j=0 b + (n -1 -j)(1 -a) j Γ(z j ) Γ(az j + b + n -1) ∆ n (z) (4.9)
and for az j + b = 0, 1, . . . , n -2,

det zj i azj +b i i,j=0,...,n-1 = n-1 j=0 b -(n -1 -j)(1 -a) j [az j + b] n-1 ∆ n (z) . (4.10)
Proof. Consequences of lemma 3.

In the special case z j := c + j, a = 1 and d := b + c with c = 0, -1, . . ., one gets [START_REF] Mehta | [END_REF] det

Γ(c + i + j) Γ(d + i + j) i,j=0,...,n-1 = n-1 j=0 j! (d -c) j Γ(c + j) Γ(d + n -1 + j) (4.11)
where n-1 j=0 (d -c) j = n-1 j=0 (d -c + j) n-1-j .

Corollary 7.

With n a positive integer and z j = 0, -1, . . .,

det Γ(z j -i) i,j=0,...,n-1 = (-1) n(n-1)/2 n-1 j=0 Γ(z j -n + 1) ∆ n (z) . (4.12)
Proof. Consequence of corollary 1.

Corollary 8.

With n a positive integer,

det 1 Γ(z j -i) i,j=0,...,n-1 = 1 n-1 j=0 Γ(z j ) ∆ n (z) . (4.13)
Proof. Consequence of corollary 2.

Corollary 9.

With n a positive integer and

az j + b = n -1, n -2, . . ., det Γ(az j + b -i) Γ(z j -i) i,j=0,...,n-1 = n-1 j=0 Γ(az j + b -n + 1) b + 1 + (n -j)(a -1) -j Γ(z j ) ∆ n (z) . (4.14)
Proof. Consequence of corollary 3.

In the special case z j := c + j and a = 1 with

d := b + c = n -1, n -2, . . ., one gets det Γ(d + j -i) Γ(c + j -i) i,j=0,...,n-1 = n-1 j=0 j! [d -c] j Γ(d -n + 1 + j) Γ(c + j) . (4.15)
Corollary 10. With n a positive integer,

det Γ(z i + w j + n -1) Γ(z i + w j ) i,j=0,...,n-1 = (-1) n(n-1)/2 (n -1)! n n-1 j=0 j! 2 ∆ n (z) ∆ n (w) (4.16) det z i + w j n -1 i,j=0,...,n-1 = (-1) n(n-1)/2 n-1 j=0 j! 2 ∆ n (z) ∆ n (w) . (4.17)
Proof. Consequences of lemma 4.

Some examples of applications

Let us sketch some examples of applications which motivated this work, i.e. the calculation of the probability density of the determinant (PDD) of random matrices. Three ensembles of n × n random matrices, with n = 1, 2, . . ., have been extensively investigated, namely the orthogonal (β = 1), unitary (β = 2) and symplectic (β = 4) ensembles of, respectively, real symmetric, complex Hermitian and real quaternion selfdual matrices [START_REF] See | Random Matrices[END_REF]. Then, the probability density of the eigenvalues x := {x j real ∈ D, j = 0, . . . , n -1} reads

P n,β (x) = C n,β ∆ n (x) β n-1 j=0 w(x j ) (5.1)
where C n,β is the normalization constant, ∆ n (x) is defined by equation (B.1) and w(x) is a non-negative weight function. Quantities one computes in random matrix theory are often expressed in terms of determinants (or Pfaffians). This is the case for the expectation value of any factorized function of the eigenvalues, Φ(x) := n-1 j=0 ϕ(x j ) [START_REF] Mehta | Probability density of the determinant of a random Hermitian matrix[END_REF][START_REF] Normand | Probability density of the determinant of some random matrix ensembles preprint[END_REF]. Let us show here this result only in the simplest case β = 2, namely with dµ(x) := w(x) ϕ(x) dx, one has

Φ := D dµ(x 0 ) • • • D dµ(x n-1 ) ∆ n (x) 2 = n! det Φ j,k j,k=0,...,n-1 (5.2) Φ j,k := D dµ(x) P j (x) Q k (x) (5.3)
where P j (resp. Q k ) is any monic polynomial (i.e. the coefficient of its highest power is one) of degree j (resp. k). Indeed, from equation (C.3), each of the two factors ∆ n (x) can be expressed as a polynomial alternant and expanded as ρ∈Sn{0,...,n-1} ε(ρ)

n-1 j=0 P ρj (x j ), where ε(ρ) is the signature of the permutation ρ := {ρ 0 , . . . , ρ n-1 }. Thereby, one gets

Φ = ρ,σ∈Sn{0,...,n-1} ε(ρ) ε(σ) n-1 j=0 D dµ(x)P ρj (x) Q σj (x) = n! ρ∈Sn{0,...,n-1} ε(ρ) n-1 j=0 Φ ρj ,j (5.4)
completing the proof of equation (5.2). According to the measure dµ(x) considered, one may take advantage of the freedom of choice of the monic polynomials in order to simplify the calculations. Thus, it may be useful to choose the set of orthogonal (or skew orthogonal for β = 1 or 4) polynomials with respect to the weight w(x) [START_REF] See | Random Matrices[END_REF][START_REF] See | Matrix Theory[END_REF] 16 . For example, taking Φ as the identity operator, the result above with ϕ(x) = 1 provides a convenient way to compute the normalization constant, e.g., for β = 2

C n,2 -1 = n! n-1 j=0 ν j ν j := D dx w(x) P j (x) 2 (5.5)
where P j are the orthogonal monic polynomials for the weight w(x).

The calculation of the PDD,

g n,β (y) := D dx 0 • • • D dx n-1 P n,β (x) δ(y -x 0 • • • x n-1 ) (5.6)
of the random matrices we consider, is based on the use of the Mellin transform. Since this transformation explores a function only on the real non-negative half-axis, one needs to compute the Mellin transform of the restriction to y ≥ 0 of both the even and odd parts of the PDD, g ± n,β (y) := 1 2 (g n,β (y) ± g n,β (-y)). From equations (5.6), with s some complex number, the Mellin transform of g ± n,β (y) reads

M ± n,β (s) := ∞ 0 dy y s-1 g ± n,β (y) = 1 2 D dx 0 • • • D dx n-1 P n,β (x) n-1 j=0 ϕ ± β,s (x) (5.7) ϕ ± β,s (x) := ε ± (x) |x| s-1 ε + (x) := 1 ε -(x) := sign(x) (5.8)
namely, an expression of the type given by equations (5.2) and (5.3) when β = 2, thus (i) For the frequently used Gaussian unitary ensemble [START_REF] Mehta | Probability density of the determinant of a random Hermitian matrix[END_REF] associated with the Hermite polynomials, w(x) = exp(-x 2 ) with D = R. Choosing the polynomials P j (resp. Q k ) to be the monomial x j (resp. x k ), one finds [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 18

M ± n,2 (s) = 1 2 C n,2 n! det Φ ± j,k (s) j,k=0,...,n-1 (5.9)
Φ ± j,k (s) = ∞ -∞ dx e -x 2 ε ± (x) |x| s-1 x j+k Re s > 0 = 1 2 1 ± (-1) j+k Γ s+j+k 2 .
(5.11)

Then, the alternate elements of det Φ ± j,k (s) j,k=0,...,n-1 being zero, we can rearrange its rows and columns so as to collect the zero elements separate from the nonzero elements. Note that this checkerboard structure of the determinant is true for any w(x)ϕ(x) with a well-defined parity and a domain D symmetrical with respect to x = 0. Thus, det Φ + j,k (s) j,k=0,...,n-1 = det Φ + 2j,2k (s) j,k=0,...,[(n-1)/2] det Φ ± 2j+1,2k+1 (s) j,k=0,...,[(n-2)/2] (5.12) det Φ - j,k (s) j,k=0,...,n-1 =    (-1) n/2 det Φ - 2j,2k+1 (s) j,k=0,...,n/2 2 n even 0 n odd (5.13) 16 See, e.g., [START_REF] See | Matrix Theory[END_REF] appendix A.14. 17 See, e.g., [START_REF] See | Random Matrices[END_REF] section 19.3. 18 See, e.g., [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] (ii) For the so-called Laguerre unitary ensemble [START_REF] Normand | Probability density of the determinant of some random matrix ensembles preprint[END_REF], w(x) = x α exp(-x) with α > -1 and D = [0, ∞[. Still choosing the polynomials P j (resp. Q k ) to be the monomial x j (resp. x k ), one finds [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 19 the result being the same for ± since the spectrum is non-negative.

Φ ± j,k (s)
(iii) For the so-called Gegenbauer unitary ensemble [START_REF] Normand | Probability density of the determinant of some random matrix ensembles preprint[END_REF]: w(x) = (1 -x 2 ) λ-1/2 with λ > 1 2 and D = [-1, 1]. Note that the special case λ = 1 2 corresponds to the the so-called Legendre ensemble with w(x) = 1. Still choosing the polynomials P j (resp. Q k ) to be the monomial x j (resp. x k ), one finds [4]20

Φ ± j,k (s) = 1 -1 dx (1 -x 2 ) λ-1/2 ε ± (x) |x| s-1 x j+k Re s > 0 = 1 2 1 ± (-1) j+k Γ λ + 1 2 Γ s+j+k 2 Γ λ + s+j+k+1 2 .
(5.17)

Therefore, the equations (5.12) and (5.

j=0 j! Γ λ + 1 2 + j Γ s 2 + j Γ s+1 2 + λ + [(n -1)/2] + j . (5.18)
(iv) For the so-called Jacobi unitary ensemble [START_REF] Normand | Probability density of the determinant of some random matrix ensembles preprint[END_REF],

w(x) = (1 -x) a (1 + x) b with a > 1, b > 1 and D = [-1, 1]. For a = b = λ -1 2
, this ensemble is identical to the Gegenbauer ensemble above. For a = b, the problem is more complicated, in particular due to the fact that w(x) is no longer an even function. To illustrate the use of the formulae we derived, let us calculate only the normalization constant C n,2 . Choosing the monic polynomials P j (x) = (x -1) j and Q

k (x) = (1 + x) k , one finds from equation (5.3) with ϕ(x) = 1 Φ ± j,k (s) = 1 -1 dx (1 -x) a+j (1 + x) b+k = (-1) j 2 a+b+1+j+k Γ(a + 1 + j) Γ(b + 1 + k) Γ(a + b + 2 + j + k) . ( 5

.19)

Then, the determinant in equation (5.2) is of the type considered in corollary 5 equation (4.8), .20) This result can be checked either from equation (5.5) using the constants associated with the Jacobi polynomials [START_REF] Bateman | Higher Transcendental Functions[END_REF] 21 , or from the Selberg integral [START_REF] See | Random Matrices[END_REF] 22 . Finally, for all these unitary ensembles (except, possibly, for the currently unknown Jacobi ensemble with a = b), the Mellin transform M ± n,2 (s) appears to be a product, or a ratio of products, of gamma functions whose arguments are linear in s. Then, from the inverse Mellin transform, the PDD is expressed in terms of Meijer G-functions [START_REF] Bateman | Tables of Integral Transforms[END_REF] 23 . For the orthogonal and symplectic ensembles the expressions are more complicated [START_REF] Mehta | Probability density of the determinant of a random Hermitian matrix[END_REF][START_REF] Delannay | Distribution of the determinant of a random real-symmetric matrix from the Gaussian orthogonal ensemble[END_REF][START_REF] Normand | Probability density of the determinant of some random matrix ensembles preprint[END_REF], but we are still led to consider similar determinants. Note that, as a by-product, one gets also the non-negative integer moments of the PDD for q = 0, 1, . . ., M n,β (q) := D dy g n,β (y) y q = 1 + (-1) q M + n,β (q + 1) + 1 -(-1) q M - n,β (q + 1) .

C n,2 -1 = n! n-1 j=0 (-1) j 2 a+b+1+2j Γ(a + 1 + j) Γ(b + 1 + j) det 1 Γ(a + b + 2 + j + k) j,k=0,...,n-1 = n! 2 n(n-1)+(a+b+1)n n-1 j=0 j! Γ(a + 1 + j) Γ(b + 1 + j) Γ(a + b + n + 1 + j) . ( 5
(5.21)

In connection with quantum coherent states, Dr K A Penson brought our attention on the boson normal ordering problem, see [START_REF] Blasiak | The boson normal ordering problem and generalized Bell numbers[END_REF][START_REF] Penson | Coherent state measures and the extended Dobiński relations Symmetry and Structural Properties of Condensed Matter[END_REF][START_REF] Penson | Hierarchical Dobiński-type relations via substitution and the moment problem[END_REF] and references therein. Let a and a † be the boson annihilation and creation operators respectively, satisfying [a, a †] = 1. The normal ordering of powers of boson monomials (a †) r a s n , with n, r, s (r ≥ s) some non-negative integers involves integer sequences of numbers which are generalizations of the usual Stirling numbers of the second kind, equation (2.9), and Bell numbers, whose values they assume for r = s = 1,

(a †) r a s n := (a †) n(r-s) ns k=s S r,s (n, k) (a †) k a k B r,s (n) := ns k=s S r,s (n, k) . ( 5

.22)

A complete theory of these sequences of numbers has been worked out. In particular, the B r,s (n) can be expressed as a sum of of an infinite series of shifted factorials (generalized Dobiński formula) and moreover, can be considered as the n-th moments of a positive weight function W r,s (x) with x ≥ 0,

B r,s (n) = ∞ 0 dx x n W r,s (x) .
(5.23)

Extending n to complex values and using the inverse Mellin transform, one gets from above many solutions W r,s (x) of the Stieltjes moment problem [START_REF] Penson | Coherent state measures and the extended Dobiński relations Symmetry and Structural Properties of Condensed Matter[END_REF]. Generalizing this approach to the integer sequences arising from the normal ordering of exponentiated boson monomials, as given by equation (5.22), also provides solutions to Stieltjes moment problems. It happens that determinants of the type we evaluate are the Hankel determinants which positivity, if it can be proved, ensures the existence of the moment problem [START_REF] Penson | Hierarchical Dobiński-type relations via substitution and the moment problem[END_REF].

Let us add that the reader can find in [START_REF] Krattenthaler | Advanced determinant calculus Séminaire Lotharingien Combin[END_REF] many methods of evaluations, lists of results and a wide bibliography on the determinant calculus. Beyond the evaluation of particular determinants, we want to point out that the properties of the s-shifted factorials given in section 2 emphasize similarities and connections which exist with the power function (see another example in appendix A), thereby providing compact formulae and possibly a guide to find new relations.

More generally, following the same arguments as for equation (C.3), one can consider polynomials of the monomials introduced above (or even of any function), e.g., with λ := det c i,k i,k=0,...,n-1 and az where the last equality is due to an overall factor y -1 which remains after the derivation. When i ≤ j, integrating j -i + 1 times (y -1) i y b+i-2 in two ways and then setting y = 1 yield where the last equality is due to an overall factor x-1 which remains after the derivation over x. When i ≤ j, differentiating j times with respect to x and integrating j -i+1 times over y the expression (xy -1) i x d-c+i-2 in two ways and then setting x = 1 and y = 1 yield where B(z, w) is the beta function. Thereby, after some elementary algebra, using equations (2.24), (2.12) and (2.13), one gets

j + b nonzero, det n-1 k=0 c i,k z j az j + b k i,j=0,...,n-1 = λ b n(n-1)/2 n-1 j=0 (az j + b) n-1 ∆ n (z) . (C.

1 j=0b 2 n- 1 j=0Lemma 3 .Proof 1 .

 12131 js) s;i i,j=0,...,n-1 = (-s) n(n-1)j -1)s s;j (b + js) s;j(3.22) and, using the multiplication law (2.22), it can be shown by recurrence that for all s n-+ (j -1)s s;j (b + js) s;j = n-1 j=0 (b + js) s;n-1 (3.23) corresponding to a rearrangement of the factors. Finally, by equation (B.5), det 1 (b + js) s;i i,j=0,...,n-1 = (-1) n(n-1)/(b + js) s;n-1 ∆ n (j → b + js) (3.24) ending the proof of equation (3.16) in the special case z j := b + js. With n a positive integer, a and b some complex numbers and az j + b = 0, -s, . . . , -(n -2)s, det (z j ) s;i (az j + b) s;i i,j=0,...,n-1 = n-1 j=0 b + (n -1 -j)(1 -a)s s;j (az j + b) s;n-1 ∆ n (z) . (3.25) This formula generalizes equation (C.6). From equation (2.30), with n -1 ≥ i ≥ 0 and az + b = 0, -s, . . . , -(n -1)s, (z) s;i (az + b) s;i = 1 (az + b) s;n-1 (z) s;i (az + b + is) s;n-1-i . (3.26) When a = 1, from the binomial formula (2.41) and the multiplication law (2.22), (z) s

D

  s;n (z 0 , . . . , z n-1 ; a, b) := det M i,j i,j=0,...,n-1= (b) (1-a)s;n-1 (az 0 + b) s;n-1 n-1 j=1 z j -z 0 az j + b D s;n-1 (z 1 , . . . , z n-1 ; a, b + s) . (3.32)Iteration of this equation, down to D s;1 z n-1 ; a, b + (n -1)s = 1, ends the proof.

  can also be given for corollaries 1-3. (iii) The extensions of lemma 1, corresponding to equations (3.3) and (3.6), apply as well to lemmas 2 and 3 and to corollaries 1-3, see equation (C.7), e.g., with Π i (z) := n-1 k=0 c i,k (z) s;-k i = 0, . . . , n -1 λ := det c i,k i,k=0,...,n-1 = 0 (3.42) then det Π i (z j ) i,j=0,...,n-1 = λ det (z j ) s;-k j,k=0,...,n-1 (3.43)

. 4 )Corollary 5 .

 45 Iteration of this equation, down to D 1 (z n-1 ) = Γ(z n-1 ), ends the proof of equation (4.1).In the special case z j := b + j = 0, -1, . . ., one recovers the result already published in[START_REF] Mehta | Probability density of the determinant of a random Hermitian matrix[END_REF] 15 , det Γ(b + i + j) i,j=0,...,n-1 = With n a positive integer,

  x) ϕ ± 2,s (x) P j (x) Q k (x) .(5.10)Now, one can consider several ensembles of random matrices associated with the classical orthogonal polynomials characterized by the weight function w(x) and the domain D [14] 17 .

= ∞ 0

 0 dx x α e -x |x| s-1 x j+k = Γ(s + α + j + k) Re s > 0 (5.15)Then, again with corollary 4 equation (4.5), one obtains det Φ ± j,k (s) j,k=0,...,n-1 =

2 [

 2 13) are still satisfied and the three determinants which occur are of the type considered in corollary 6 equation (4.11), e.g., det Φ + 2j,2k (s) j,k=0,...,[(n-1)/2] = Γ λ + 1 (n-1)/2]+1 det Γ s 2 + j + k Γ s+1 2 + λ + j + k j,k=0,...,[(n-1)/2] = [(n-1)/2]

7 )D. 2 .

 72 Appendix D. Other proofs of equations(3.14),(3.21) and(3.35),(3.37) These identities can be proved by recurrence on i. Let us also give a proof which illustrates another way to handle shifted factorials, namely they can be generated by repeated derivations and/or integrations, e.g., 1) j = [b] -j . (D.2) D.1. Other proof of equation (3.14)Differentiating j times (x -1) i x b+j-1 in two ways (binomial formula and chain rule derivation of a product) x = 1, only the term with ℓ = i is nonzero. Thereby, one getsi k=0 (-1) i-k i k [b + j + k -1] j = [j] i [b + j -1] j-i (D.4)where [j] i , and thus the right-hand side, vanishes for i > j, see equation (2.10). Since from equations (2.12) and (2.13)[b + j + k -1] j = Γ(b + j) Γ(b + i) [b + i -1] i-k (b + j) k (D.5) [b + j -1] j-i = Γ(b + j) Γ(b + i) (D.6) one recovers equation (3.14) (with s = 1 for simplicity). Other proof of equation (3.21) Assume first i > j. Then, as above, one gets d dy i-j-1 (y -1) i y b+i-2 y=1 = i k=0 (-1) i-k i k [b + i + k -2] i-j-1 = 0 (D.7)

1 0D. 3 . 1 ( 1 (x - 1 )

 13111 i + k -1) j-i+1 = dy 1 (y 1 -1) i y b+i-2 i (j -i)! B(b + i -1, j + 1) (D.8)where B(z, w) is the beta function[START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 25 , thus i+ k -1) j-i+1 = (-1) i [j] i Γ(b + i -1) Γ(b + i + j) . (D.9)Now, since from equations (2.24), (2.12) and (2.13)(-1) i-k [b + i + k -2] i-j-1 = (-1) i-k (b + i + k -1) j-i+1 (D.10) = Γ(b + 2i -1) Γ(b + j) × 1 -b -2(i -1) i-k 1 (b + j) k (D.11)and furthermoreΓ(b + i -1) Γ(b + i + j) = Γ(b + 2i -1) Γ(b + j) × 1 (b + i -1) i (b + j) i (D.12)the sum over k in equation (D.7) for i > j and (D.9) for i ≤ j does correspond to the sums considered in equation (3.21) (with s = 1 for simplicity). Note that since [j] i vanishes for i > j and with equation (D.10), the relation (D.9) is true in all cases. Other proof of equations (3.35), (3.37) Assume first i > j. With now two variables x and y, as above, one gets xy -1) i x c+j-1 y d+i-2 i-k i k [c + j + k -1] j [d + i + k -2] i-j-1 i x d+i-2 x=1 = 0 (D.13)

  j + k -1] j (d + i + k -1) ) i-ℓ j ℓ [i] l [c + j -1] j-l B(d + i + ℓ -1, j -ℓ + 1) (D.14)

(- 1 )

 1 j + k -1] j (d + i + k -1) j-i+1 = [j] i Γ(c + j) Γ(c + i) Γ(d + i -1) Γ(d + i + j) i ℓ=0 i-ℓ j ℓ (d + i -1) ℓ [c + i -1] i-ℓ = Γ(d + 2i -1) Γ(d + j) Γ(c + j) Γ(c + i) × [j] i (d -c) i (d + j) i (d + i -1) i (D.15)where the last equality follows from the binomial formula (2.46). Now, since[c + j+k -1] j [d + i + k -2] i-j-1 = [c + j + k -1] j (d + i + k -1) j-i+1 (D.16) = Γ(d + 2i -1) Γ(d + j) Γ(c + j) Γ(c + i) × [c + i -1] i-k [d + 2(i -1)] i-k (c + j) k (d + j) k (D.17) the sum over k in equation (D.13) for i > j and (D.15) for i ≤ j does correspond to the sum over k in equation (3.35) (with s = 1 for simplicity). Note that since [j] i vanishes for i > j and with equation (D.16), the relation (D.15) is true in all cases.

  .11) Then, for s nonzero, values of (ks) s;n follow from equation (2.7), in particular (s) s;n = n!s n .2.3. Relations with the gamma function and definition of the generalized s-shifted factorialOne has[START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 5

  For a proper choice of determination the power function fulfils (z t ) r = z tr . No equivalent general relation exists for the s-shifted factorial. Although (z -1 ) p has no simple relation with ((z) p ) -1 , let us point out the following expression for any integers n ≥ p ≥ 0, by equations(2.

								24), (2.22) and (2.6), with
	z = 0, -s, . . . , -(n -1)s,
	1 (z) s;p	= (z + ps) s;-p =	(z + ps) s;n-p (z) s;n	=	z + (n -1)s -s;n-p (z) s;n	(2.30)
	recovering for s = 0 the relation (z p ) -1 = z -p = z n-p (z n ) -1 .
	2.5. Generalized Pascal triangle property and s-difference operator
	The multiplication law (2.22) and equation (2.18) yield
		(z) s;t -(z -s) s;t = ts (z) s;t-1	(2.31)
	which generalizes the Pascal triangle property for binomial coefficients, by equation (2.12),
		z + 1 n	=	z n	+	z n -1	.	(2.32)
	Let us define the s-difference operator ∆ s on functions f of z by
		∆ s f(z) := f(z + s) -f(z)	(2.33)
	(this operator must not be confused with the product of differences ∆ n (z) introduced latter in section 3 and
	defined by equation (B.1)). It follows immediately from equation (2.31),
	∆ s (z) s;t = ts (z + s) s;t-1				∆ -s (z) s;t = -ts (z) s;t-1	(2.34)
	and iterating these formulae, e.g., the first one
		∆ p s (z) s;t = [t] p s p (z + ps) s;t-p	(2.35)
	recovering for s = 0 the expression of d p dz p z n .
	2.6. Generating function and binomial formulae
	With x some complex variable, let G s;z (x) be the generating function of the s-shifted factorials (z) s;n ,
		G s;z (x) :=	∞ n=0 (z) s;n	x n n!	, |sx| < 1	(2.36)
	and, using equation (2.7),
		G s;z (x) = G 1; z s (sx) .	(2.37)
								that, based on the
	reflection formula and the Gauss' multiplication formula, 2 sin(πkz) follows the known multiplication law
	similar to equation (2.28) [5] 9 ,
		2 sin(πkz) =	k-1 ℓ=0	2 sin π z +	ℓ k	.	(2.29)

  6.1.1. where [x] denotes the largest integer less than or equal to x. From equation (5.11), the three determinants above are of the type considered in corollary 4, equation (4.5), e.g.,

	[(n-1)/2]
	det Φ + 2j,2k (s) j,k=0,...,[(n-1)/2] = det Γ s 2 + j + k j,k=0,...,[(n-1)/2] =	j! Γ s 2 + j .	(5.14)
	j=0

See, e.g.,[START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] 6.1.20.

See, e.g.,[START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF] 1.392 (1.).

See[START_REF] Mehta | Probability density of the determinant of a random Hermitian matrix[END_REF] equation (A.12).

See, e.g., [4] 6.1.1.

See, e.g., [4] 6.2.1 and 6.2.2.

See, e.g.,[START_REF] Bateman | Higher Transcendental Functions[END_REF] taking h j from equation 10.8(4) and k j from equation 10.8[START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF], then ν j = h j /k 2 j .

See, e.g.,[START_REF] See | Random Matrices[END_REF] section 17.6.

See, e.g.,[START_REF] Bateman | Tables of Integral Transforms[END_REF] section 7.3 (43).

See, e.g., [4] 6.2.1 and 6.2.2.

Acknowledgments

This paper was originally motivated by the evaluation of some determinants with gamma functions as elements which occur in works done in collaboration with M L Mehta. P Moussa drew our attention to the importance of the exponential character in z of the generating function of shifted factorials in connection with the binomial formula; we had several stimulating discussions on the subject. We are also grateful to both of them for critically reading the manuscript. Finally, we thank the referees for asking us to add some examples of applications.

Appendix A. Finite sum of s-shifted factorials of arithmetic progression

For p a non-negative integer and a, r and s some complex numbers, we compute the finite sum of s-shifted factorials of arithmetic progression to n terms, using the same trick as for the sum of powers of natural numbers. By the binomial formula (2.41),

Summing up both sides of this equation for k = 0, . . . , n -1 yields the recurrence formula on p, for n fixed,

The first two sums are independent of s,

, and with s nonzero, an explicit expression of S s;p,n (a, s) can be obtained directly from the generalized Pascal triangle property (2.31),

where z -1 = a -s. This result can be checked by recurrence using the general equation (A.3). Similarly, for r = -s one gets

Thus, for a = r = s = 1 one has, respectively, for the rising and the falling factorials

Further general properties follow from equations (2.5) and (2.7):

S s;p,n (-a, -r) = (-1) p S -s;p,n (a, r) (A.9)

Appendix B. Product of differences

With the notations of equation (3.1), the product of differences ∆ n (z) is defined by

The following relations are immediately obtained with a and b some complex numbers:

Finally, with a and b some complex numbers, in the special case z j := b + aj, the product of differences reads

Appendix C. Vandermonde's determinant

It is well known [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF][START_REF] See | Matrix Theory[END_REF] 24 that the Vandermonde determinant det (z j ) i i,j=0,...,n-1 is equal to the product of differences defined by equation (B.1), namely,

More generally, let us consider any set of n linearly independent polynomials in z each of degree less than n,

Then, since the determinant of the product is the product of the determinants, one gets for the polynomial alternant det p i (z j ) i,j=0,...,n-1 = det c i,k i,k=0,...,n-1 det (z j ) k j,k=0,...,n-1

Choosing the p i 's to be monic polynomials of degree i (e.g., the monomials z i ), then c i,k = 0 for k = i + 1, . . . , n -1 and c i,i = 1, therefore λ = 1 in equation (C.2). Now, with b i some complex numbers, it follows from the binomial formula that (z + b i ) i is an other choice of monic polynomial of degree i, hence

When b i = b, the relation above is also a direct consequence of equations (C.1) and (B.2). Since (z i ) -1 = (z -1 ) i , with a and b some complex numbers, one immediately obtains from equations (C.1), (B.3) and (B. ∆ n (z) az j + b = 0 . (C.6) 24 See, e.g., [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF] section 14.311 or [START_REF] See | Matrix Theory[END_REF] section 7.1.