CC BY
15DOI: 10.15393/j3.art.2019.6830
UDC 517.53
V. G. Lysov
INTEGRAL REPRESENTATIONS FOR THE JACOBI - PINEIRO POLYNOMIALS AND THE FUNCTIONS OF THE SECOND KIND
Abstract. We consider the Hermite-Pade approximants for the Cauchy transforms of the Jacobi weights in one interval. The denominators of the approximants are known as Jacobi - Pineiro polynomials. These polynomials, together with the functions of the second kind, satisfy a generalized hypergeometric differential equation. In the case of the two weights, we construct the basis of the solutions of this ODE with elements of different growth rate. We obtain the integral representations for the basis elements.
Keywords: Hermite-Pade approximants, Jacobi- Pineiro multiple orthogonal polynomials, functions of the second kind, integral representations, generalized hypergeometric functions 2010 Mathematical Subject Classification: 33C20, 33C45
1. Introduction. We start from the definition of the Hermite-Pade approximants. Consider a vector / := (/1,...,/r) of germs of analytic functions in a neighborhood of infinity:
/j(z) = E PP j = 1.....r.
k=0
Hermite-Pade approximants of type II for a vector / and a multi-index
n := (n1,..., nr) E Z+ are the rational functions (^Qr,...,"Qfr) , such
that the degree of the denominator Qn does not exceed |n| := n1 + ■ ■ ■ + nr and the following interpolation conditions at infinity hold:
Rnj (z):=(Qn/j - Prtj )(z) = oj , z j = 1,... ,r. (1)
© Petrozavodsk State University, 2019
The conditions (1) form a linear homogeneous system of |n| equations for |n| + 1 unknown coefficients of Qn. So, a nontrivial solution always exists. The numerator P,,j is the polynomial part of the Laurent expansion of Qnfj at infinity. The remainder function R,,j is the main part of the same Laurent expansion. It is called the function of the second kind.
In this paper, we consider the Hermite-Pade approximants for a set of Markov functions:
where A := [0,1] and w\,..., wr are the Jacobi positive weights in (0,1):
wj (x) := xa (1 - xf, x G (0,1), aj $ > -1, j = 1,...,r. (3)
It is well known (see [10]), that for the Markov functions (2) the interpolation conditions (1) are equivalent to the system of orthogonality relations with respect to the weights Wj :
/a,«.*wj(x)dx = °, t = °,...,nj-1, j = i,...,, (4)
Due to the orthogonality conditions (4), the left-hand side of (5) does not depend on the choice of the polynomial pnj of degree not exceeding nj.
It is natural to assume that the exponents aj have pairwise different fractional parts: aj — ak ^ Z as j = k. Then the set of weights (3) forms an algebraic Chebyshev system (AT-system, see [10]) in A. It follows that the polynomial Qn has |n| simple zeros in A. Thus, all the indices n are normal, i.e, the degree of Qn is equal to |n|, it is determined up to a constant factor, and the Hermite-Pade approximants are uniquely determined.
The Hermite-Pade approximants for the set (2), (3) were introduced in [11]. The polynomials Qn are called the Jacobi-Pineiro polynomials of multiple orthogonality [2]. Whenever it is necessary to emphasize the
A
The functions of the second kind can be represented as follows:
A
dependence on the parameters a and the denominators Qn and the functions of the second kind Rnj will be also denoted by QOf and R^j • In the case r =1, Q^ is the classical Jacobi polynomial on [0,1] and R^ is the corresponding function of the second kind, see [13], [9].
The Jacobi - Pineiro polynomials have a number of applications. E. Mu-khin and A. Varchenko [7] show that the zeros of Qn along with the zeros of their Wronskian-type determinants are the unique solution of a certain Bethe Ansatz equation. C. Smet and W. Van Assche [12] show that their Mellin transforms can be applied to prove the Aperi theorem on the irrationality of Z(3) and the Ball-Rivoal theorem on the infinite number of irrational points among {Z(2m + 1)}. M. Adler, P. van Moerbeke and D. Wang [1] relate the Jacobi-Pineiro polynomials to certain problems of random matrix minor processes in the percolation theory.
The algebraic properties of Qn were studied in [2], [4]. The limit zero distribution of Qn was investigated in [8], the asymptotics near the edges of A was considered in [3]. The aim of this paper is to obtain integral representations, which can be used for describing the asymptotic behavior of Qn and Rn,j in different asymptotic regimes that are important in applications.
The Jacobi - Pineiro polynomials have an explicit representation [2,11]:
_a • .Jn' \
n V dzj (1 - z r|n|. (6)
The operators in the product of the formula (6) commute. The orthogonality relations (4) can be easily checked using the integration by parts. Similarly to the classical case r = 1, we will call the representation (6) the Rodrigues formula. The formula (6) fixes a normalization of Qn:
Qn(0) = n ^ + ^
j=1
n
where (a)n := (a) ■ ■ ■ (a + n — 1) is the Pochhammer symbol. In what follows, we will use this normalization.
2. Generalized hypergeometric differential equation. The
Jacobi - Pineiro polynomials preserve many properties of classical Jacobi polynomials: they have Rodrigues representations, satisfy certain recurrence relations. Another such property is the relationship with (generali-
zed) hypergeometric functions (see [4]):
(1 - = -FT ( a + ' + * — 0^.z) = : F0(z), (7)
where 1 = (1,..., 1) and
F ( a A = ^ (ai)fc ... (aP)k z^
V& ; Z) := k= (bi)k ... (bq )k k! .
It immediately follows from (7) that the function y(z) = (1 — zQn(z) satisfies the linear ordinary differential equation of order r + 1:
= 0, Lz := (z|z — 0 —lnl)ll (zdzz + a + 1 + j —
j=i
- dz I! (zdz+a
3 = 1
The set of functions
Fo, z-aiFi,..., z-aFr forms a basis of the solutions of (8), where
F/a + n +(1 — aj)1, —0 — aj — |n| . z Fj(z):=r+lM [a]j + (1 — a)[1]j, 1 — a 'z
and [ a ]j := (a1,..., aj-1, aj+1,..., ar) is the projection of a onto the hyperspace orthogonal to ej.
On the other hand, it can be shown that for some constants bj = 0 we have
Rh,3 (z) _ n
-(-z)a Fo(z) + bj Fj (z).
Vn(U) sm n«j
Therefore,
ji
(1 - zf Qn, z-aiRn,i,..., z-arRn,r (10)
is another basis in the space of solutions of (8).
3. Integral representations. Further, for the sake of simplicity, we restrict ourselves to the case of r = 2 weights.
For large values of |n|, the basis (10) is preferable to the basis (9), since only the first element of (10) has the maximum growth rate. It would be helpful to construct a basis where all the elements have pairwise different growth rates. Nonetheless, it turns out that for ai — a2 E (0,1) and for the diagonal indices n = n2 the functions z-ai Rn ,i and z-a2 Rn ,2 are asymptotically close. Indeed, in [6] it is shown that Rn2 ^ z"2-ai for z E C \ (—w, 1] as n1 = n2 ^ w, where za2-ai is analytic outside (—to, 0] and positive for z > 0.
Therefore, along with the functions Rn,j, we consider the functions
Rj (see [5]):
Rh := —za2-ai:= za2-ai— Rn,2. (11)
Thus, we establish that
(1 — z)^Qn, z-aiRn,i, z-aiR^ (12)
is a basis of the solutions of (8) for r = 2.
It directly follows from the definition (11) that the functions Rn,j are analytic outside (—to, 1]. Applying the Sokhotski-Plemelj formula to (5), we calculate the jumps of Rn,j in the interval (0,1):
Rn,j(x + i0) — Rnj(x — i0) = —2niQn(x)wj(x), x E (0,1).
Clearly, the jumps of Rn j in (0,1) are equal to zero and the singularities at the point 1 are removable. So,
in fact, Rn j
are analytic outside (—to, 0]. For a E (0,1) the following representation for z-a holds:
0
a sinna f (—x)-adx ^ . .
z-a =- ----, z E C \ (—w, 0].
n I z — x
—oo
So for ai — a2 E (0,1) we have the following integral representation for
the function Rn2:
0
Rn 2(z) = Sinn(ai — ^ / x)«2-«idx. (13)
' n ./ z — x
— oo
The formula (13) carries out an analytic continuation of Rn 2 from the domain C \ (—w, 1] to C \ (—w, 0].
We introduce the notation for three double integrals:
r £«,i+ai —a2dt f s™2+a2 (1 _ s)™i+™2+^ds
/ tni+ai—a2dt / Fn0(Z) f f
Fn,i(z) :
_ z)«+1 / (s _ t)
«2 +1
Yz Yt
1
r t«i+ai —a2dt f s«2+a2 (1 _ s)™i+™2+^ds
F«, i(z)
J (t _ z)ni+1/ (s _ t)n2+1
0 Yt
0 1
f tni (_t)ai—a2dt f sn2+a2 (1 _ s)ni+n2+^ds J (t _ z)ni+W (s _ t)n2+1 ;
where Yz, Yt are small enough positively oriented circles around the points z and t, respectively. The first two integrals absolutely converge under standard conditions on the parameters: a1;a2> _1, n1;n2 G Z+. For absolute convergence of the third integral, we additionally require that a1 _ a2 G (_n1 _ 1,n2 + 1). Under these conditions, these integrals determine holomorphic functions in the following domains:
F«,o G H (C \ R\A) , Fn,1 G H (C \ A), F«,1 G H (C \ (_rc,0]).
The relation between these integrals and the considered Hermite - Pade approximants is given in the following theorem.
Theorem 1. We have the following integral representations for the set of functions (w1Qn, Rn,1; R« 1):
/F«0 Fni sinn(a1 _ a2) F«-A
(w1Qn.R« 1, an 1) = _ (^, ^, 1 n ' ). <14)
The integral representations for Q«, R«, 1 are simple consequences of the Rodrigues formula (6), the formula for R« 1 is less trivial.
4. Integral representation for Q«. Rewrite the Rodrigues formula (6) for the case r = 2:
1 d« 1 d«
zai(1 _ z)^Q«(z) = — -—zrai+ai —a2 —-—zra2+a2(1 _ z)rai+ra2+^. v n1! dz"4 n2! dz«2 v '
We obtain the integral representation for Q« from this formula by applying twice the Cauchy formula for derivative of an analytic function. Indeed, according to the Cauchy formula, we have
1 d"2
n2! dt"2 ( )
1 f sn2+a2 (1 - s)"1 +ra2+^ds G(t)
2ni / (s - t)n2+1 ' 2ni
it
:i5)
By definition, the functions F",0, are related to G by the equalities:
i
^ , , rtni+ai-«2G(t)dt ^ , , /•tni+«1-«2G(t)dt , N
F«W = / (t - z)G" , *»<*> = / (t -z^ . (16)
Yz o
Applying the Cauchy formula again, we obtain the required:
Wl(z)Qn(z) = — — — Z"1+a1 -2 G(z) = -±- I tn1+"1-"2 G(t)dt
Wl(z)Qn(z) = 2ni ni! dz"1 Z G(Z) = (2ni)^ (t - z)"1+i •
lz
Note that by the formula (6) (for r =1) the function G defined by the formula (15) is expressed through the classical Jacobi polynomials:
Gt) = ta2 (1 - t)n1+^Q^2^(t). (17)
5. Integral representation for Rn,1. We need the following lemma, which is a consequence of the partial integration formula.
Lemma 1. 1) Let u,v G Cn[0,1] and u(x) := xc0(1 - x)c1 u(x), v(x) := xdo(1 - x)d1 u(x), where c0 + d0 > n - 1 and c1 + d1 > n - 1; then
1 1
/dnv f drau
u(x)dX"(x)dx = (-1) ; dX"(x)v(x)dx
oo
2) Let u, u G C"[1, + to), u(k)(x), u(k)(x) = O (x-k) as x ^ +to, k = = 0,... , n and u(x) := xc0-c1 (x - 1)c1 u(x), v(x) := xdo-d1 (x - 1)d1 u(x), where c1 + d1 > n - 1 and c0 + d0 < n - 1; then
/drav f drau
u(x)dx"(x)dx = (-1) y dx"(x)v(x)dx 11
The conditions of the lemma are sufficient for the absolute convergence of the integrals and equality to zero of all non-integral terms, see section 7 for the proof.
Integral representation for Rn,i is derived from (5) and (16) applying Lemma 1 and (17):
i
1 (Z) 'J z-t = 0
1 1
= JL Í JL 1_dl tm+a1-«2G(t) = -1 /trai+ai-a2G(t)dt
2nzjz -1 ni! dt™ () 2nzJ (t - z)ni+1 '
00
We obtain one more integral representation for Rn,1. We use the formula (15) and again apply Lemma 1 to continue the chain of identities:
1
/trai+«i-«2 dt 1 d™2
t-A t™2+a2 (! - t)-i+n2+^ =
(t - z)ni+1 n2! dt™ v ;
0
f1 1 d™2 t™i+ai-a2
= (-1)n2/ tn2+a2 (1 - t)ni+n2+^ dt- " '
l0 n2 ! dtn2 (t - z)ni+1
(_ 1)ni+n2 + 1 1 1 dn2 ( t Ani+ai-a2
A-)— tn2+a2(1 - t)ni+n2+^dt^ , " -+, '
zn2+1-«i+«2 ^ J n2! d(t/z)n2 i\ni+1
n2! d(t/z)n2 fx o v
Denote by qn a polynomial of degree nr defined by the formula:
(1 _ +"2+1 d™2 xrai+ai-a2
qn(x) := ----i--;-JZ-^rr. (18)
w xai-a2n2! dx"2 (1 - x)ni+r v 7
Then we obtain the following relation for Rn,r:
r1 znidt
- Rn,r(z) = J tn2+ai(i - tr+n2+^(t-i. (19)
6. Integral representation for Rnr. We now turn to the proof of the last relation (14). The definition of Rn,j (see (11)) implies that R" r = — R;2. We will use the representation (13), which is valid under the condition ar — a2 G (0,1). First we prove what is required under this
restriction, and then remove the restriction by applying the principle of analytic continuation on a. So, we have the following chain of equalities:
0
nR^z) f Rn)i(x)(-x)a2-aidx
sin n(a1 — a2) J z — x
—oo
( x)a2 aidxf ra2+ain _ oNn1+n2+^ qn (-) ds
S«2+ai (1 _ s)ral+ra2+^ x qn
J x — z J ( ) (s — x)n +n2+1
0
1 0
f + , s + /" xniqn (-) (—x)a2-aidx
Sra2+ai (1 _ s)ni+"-2+Pds / _ =
J ( ) J (s — x) ni +n2+1(x — z)
0 -œ
1
= / sn2+ai (1 — s)ni+n2+^I(s,z)ds, (20)
where
0 / )
, , f x"1 q" (-x)a2-a1 dx , , n
I(s,z):=/ . ^;+ +1.--, s> 0, z G C \ (-to, 0].
V 7 J (s - x)"1+"2+1(x - z) ' \ v J
Let us transform the integral I(s,z). First, we change the variable x to t = s/x:
0 /-)". 1 , , / s )a2— «1 s
.. . , (?)ni qn(t) T2 dT
1 (s,z) I , „N ni +T12+1
s - z) ' ' ' lé - z
0
(—1)ni+n2+W Tn2qn(t)(—r)ai-«2dT
zs"2 +«1— «2 J (1 - t)"1+"2 + 1 - '
— TO
The expression (18) for the polynomial q" can be rewritten, changing the branch of the power function in the numerator and denominator:
(1 - x)ra1+ra2 + 1 d"2 x" 1 (-x)"1—"2 q"(x) = (—x)a1—a2n2! dx"2 (1 - x)"1+1 •
We substitute this expression for q" in the integral and again apply Lemma 1 (recall that for now a1 - a2 G (0,1)):
o
( (-1)ni+n2+r Tn2dr 1 d"2 Tni(-t)
1 (s, z)
zsn +«i-«^ (z - t) n2! dT"2 (1 - t)ni+r
-<x
0
(-1)ni+r f Tni (-T)ai-a2dT 1 d"2 t"
zsra2+«i-a2 J (1 - t)"i+r n2! dT"2 - T) '
Note that the following identity holds:
1 d"2 t"2 (f)
"2
Z J
"2 +1
n2! dT"2 (f - t) (f - t)
Using this equality, we remove one floor in the formula for I(s,z). Then the substitution t = t gives:
(-1)"i+r ? t"i(—t)"i-"2 (f)"2 dT _ (S,Z) = zS"2+«i-«2 J (1 - T)"i+1 ( s _ T)"2+1 =
tni (-t)ai-a2 dt
= (zs)
a2 - ai
zR_
(t - z)"ir1(s - t)"2+r
In the last integral, we deform the contour by replacing the ray zR- with R-. This can be done by the Cauchy theorem, since the singularities of the integrand at the points s and z are in one of the two angles formed by the rays, and it has the following behavior in a neighborhood of infinity: O (r/tn2+2+a2-ai) = o (r/t). We get the following expression for I(s,z):
o
/t"i ( —t)ai-«2dt
, t,(+),—d"^. (t - z)"i + r(s - t)"2 + r
-TO
Substituting this formula in (20), we obtain the expression for R matching the required after changing the order of integration:
-nR",i(z) nR",2(z)zai-a2
* ",2
sin n(ar - a2) sin n(ar - a2)
1 o
=1 ^(1 - sr+..,+«dsj =^
ai — a2
Thus, the following implication is proved:
n _, T?w ^ sin - Q^* ( S
ai - a2 e (0,1) Rn1(z) =-Fn1(z).
n
Let us note that in the above identity the left-hand side Rn, 1 := WW1 Rn,2 — Rn, 1 analytically depends on a in the domain {a : Re aj > — 1, j = 1, 2} by virtue of the proved integral representations for Rn , j (the integral for Rn ,2 is obtained from the integral for Rn , 1 by permutation of indices 1 o 2). The integral Fn 1 in the right-hand side of this equality is an analytical function in {a : Re (a1 — a2) e (—n1 — 1, n2 + 1), Re aj > —1, j = 1, 2}. By the uniqueness theorem, the equality holds in the whole domain.
Note that for the function Fn1, the analogue of representation (16) holds:
0
(z)= itni(—t)"1-"2g(^dt (2i)
Fn,1(z) = y (t — z)ni+1 , (21)
—TO
where g is the function of the second kind for the classical Jacobi polynomials (compare with the expression (17) for G):
i ^
sn2 +«2 _ s)nl+n2+ßds _ _G(¿)
(s - i)™2+1 = 2ni "n2
g(t) := S-((1 Sn+i dS = = _Rn2'ni+ß(t). (22)
7. Proof of Lemma 1. 1. It is enough to verify the convergence of at least one of the integrals and equality to zero of all non-integral terms in the formula of integration by parts:
n— 1
(Wn)) (x)dx = J](_1)fc (u(fc)v(n-1-
fc=Q
+ (_1)n / (u(n)v) (x)d
u (x)dx.
The singularities at the ends are integrable, because
(u(n)v) (x) = xc0+d0-n(1 - x)Cl+dl-raUIra(x)
for certain function ttn G C[0,1] and Cj + dj — n > -1, j = 0,1. All the non-integral terms equal zero, because
(u(k)v(n-1-k)) (x) = xc0+d0-n+1(1 — x)Cl+dl-n+1Mfc(x)
1
1
1
Q
for certain function ttk G Cmin(k+1,ra—k) [0,1], where k = 0,..., n _ 1.
2. We apply the formula of integration by parts. Similarly to the previous case, we verify that the integrals converge at point 1 and all the non-integral terms at this point are equal to zero. We only need to verify the convergence and vanishing of the non-integral terms at infinity. By the Leibniz formula, it is easy to see that for k = 0,..., n we have
dk / 1N Ci
—r(1--) w(x) = O (x—k) , x ^
dxk V x/ w v '
By the Leibniz formula for the kth derivative of the product:
k
u(k)(x) = xC0—k ¿ bk,j x^-j (1 _ -) Ci w(x) = O (xC0—k) , x ^
dx x
j=o
where bkj := (k)(k _ j)!(kC0j are constants. Similar relations hold for the function v. Eventually, we get that the )unction u (ra)v is integrable at infinity, because (u (ra)v) (x) = O (xc°+d°—«= O (x—1—£) as x ^ where e := n _ 1 _ c0 _ d0 > 0 and all the non-integral terms vanish at infinity:
(u(k)v(ra—1—k)) (x) = O (x—£) , x ^ k = 0,..., n _ 1.
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Received August 16, 2019. In revised form,, October 29, 2019. Accepted October 30, 2019. Published online November 5, 2019.
Keldysh Institute of Applied Mathemtics, RAS 4 Miusskaya sq., Moscow 125047, Russia E-mail: [email protected]
