Abstract
In this work, we solve the general linearization problem for the generalized Bessel polynomials using their inversion formula. For some particular values, we get a recurrence relation satisfied by the linearization coefficients from which we deduce their nonnegativity. We also recover a result given by Berg and Vignat (Constr Approx 27:15–32, 2008) and derived an explicit formula that generalizes a result by Atia and Zeng (Ramanujan J 28:211–221, 2012).
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1 Introduction
Given a sequence of polynomials \(\{p_n(x)\}_{n\in {\mathbb {N}}_0}\), one may like to know something about the nonnegativity of the coefficients \(\displaystyle {L_{k}(m,n)}\) in
Equation (1) is called the linearization formula of the polynomial sequence \(\{p_n(x)\}_{n\in {\mathbb {N}}_0}\) and \(\displaystyle {L_{k}(m,n)}\) the linearization coefficients. It is often important to know whether the linearization coefficients are positive or non-negative (see e.g. [5, 9, 13, Chap. 9 and references therein], [14, 18]). In the sequel, we use the generalized hypergeometric series defined by
where \((a)_n\) denotes the Pochhammer symbol (or shifted factorial) given by
Consider the generalized Bessel polynomials defined (in 1949 by Krall and Frink [19]) for \(N\in {\mathbb {N}}\) by (see also [12, Chap. 2], [13, p. 123], [17, Section 9.13, p. 244])
and (see [12, p. 13], [13, p. 123], [17, Remarks, p. 246])
We refer to [12, 13, 17, 19] concerning references to the literature and the history about Bessel polynomials.
In this work, we give the linearization formulae for the generalized Bessel polynomials \((y_n(x;\alpha ))_n\) and \((\theta _n(x;\alpha ,\beta ))_n\); we find recurrence relations satisfied by the linearization coefficients of the polynomial family \((\theta _n(x;0,\beta ))_n\) (from which their nonnegativity is deduced) and the polynomial family \((y_n(x;\alpha ))_n\). For \(\beta =2\) in \((\theta _n(x;0,\beta ))_n\), we recover results given by Berg and Vignat [5] and we derived an explicit formula that generalizes a result by Atia and Zeng [3]. Following the work by Atia and Zeng [3], we simplify the linearization coefficients of the polynomial family \((\theta _n(x;0,\beta ))_n\) from a double sum to a single sum.
2 Linearization formulae of the Bessel polynomials
In this section, we solve the more general linearization problem
for the generalized Bessel polynomials. The linearization formula (4) follows from the hypergeometric representation of the generalized Bessel polynomials and their inversion formula, that is, a formula expanding the basis \((x^n)_{n}\) into a family of polynomials \((p_n(x))_n\) with \(\deg p_n =n\)
The inversion formula of the generalized Bessel polynomials \((y_n(x;\alpha ))_n\) is given by (see e.g. [7, 12, p. 73], [15, 23, Eq. (7), p. 294], [24, 27])
The polynomial \(\theta _n(x;\alpha ,\beta )\) is solution of the differential equation (see [12, Eq. (29), p. 13], compare [13, p. 123])
Equation (6) follows immediately from the differential equation (see e.g. [12, Eq. (26), p. 12], [17, p. 245], [19, Eq. (33)])
satisfied by \(y_n(x;\alpha )\) and the relation \(\theta _n(x;\alpha ,\beta )=x^ny_n(2(\beta x)^{-1};\alpha )\). In [12, Eq. (5), p. 42], the so-called pseudo-generating function for \(\theta _n(x;\alpha ,\beta )\) is given as follows:
From this pseudo-generating function, we prove that
Proposition 1
The inversion formula of the Bessel polynomials \((\theta _n(x;\alpha ,\beta ))_n\) is given by
Proof
From (7), we get
Since
and
it follows that
Therefore
From the relation [23, Eq. (1), p. 56]
we deduce by equating the coefficients of \(u^n\) in the latter expression that
from which the result follows. \(\square \)
Remark 2
For \(\alpha =0\) and \(\beta =2\), we have the Bessel polynomials (see [5, 12])
and (8) becomes (after multiplication and division by n!)
Since \((-m)_{n-m-1}=0\) if \(n-m-1>m\), i.e., if \(0\le m < \frac{n-1}{2}\), and \(\frac{(-n)_m(-m)_{n-m-1} }{(-1)^{m+1} m!n! }=\frac{(-1)^{n-m}}{(n-m)!(2m+1-n)!}\), the above inversion formula coincides with the one due to Carlitz [6] (see [3, Eq. (13)], [5, Eq. (17)], [12, p. 73])
To solve (4), we proceed in general as follows (see [1, 15]): If
then by the Cauchy product,
with
Combining the preceding result with the inversion formula
we get
It follows then from the representations (2) and (3) of the generalized Bessel polynomials and their inversion formulae (5) and (8) that
Theorem 3
The linearization formula for the generalized Bessel polynomials
has the coefficients
The linearization formula for the Bessel polynomials
has the coefficients
In the above theorem, \((a_1,a_2,\ldots ,a_k)_n=(a_1)_n(a_2)_n\cdots (a_k)_n\).
2.1 Special cases
For \(m=0\) and \(\alpha =\gamma \) in (10), we get the multiplication formula (see e.g. [8, 20, 25])
and for \(m=0,\, \lambda =\alpha ,\, \delta =\beta \), we deduce from (11) the multiplication formula
3 Recurrence equations and nonnegativity of the linearization coefficients
In what follows, we derive a recurrence equation for the linearization coefficients \(L^{(\beta )}_k(m,n,a,1-a)\) of the linearization formula
in the specific case \(\alpha =0\), \(b=1-a\). Moreover, we give conditions for those coefficients to be nonnegative and we recover the result by Berg and Vignat [5] as a particular case. We also derive a mixed recurrence relation satisfied by the linearization coefficients \(L_k^{(\alpha )}(m,n,a,b)\) of the linearization formula
Proposition 4
The polynomials \(\theta _n(x;\alpha ,\beta )\) and \(y_n(x;\alpha )\) satisfy, respectively, the structure relations (compare [13, Eq. (4.10.12)])
Proof
Substitute \(\theta _n(x;\alpha ,\beta )=A_nx^n+B_{n}x^{n-1}+C_nx^{n-2}+\ldots \) and \(y_n(x;\alpha )=A'_nx^n+B'_nx^{n-1}+C'_nx^{n-2}+\cdots \), respectively, in the structure relations
and equate the coefficients of \(x^n, x^{n-1}\) and \(x^{n-2}\) to get the coefficients \(c_n,\, d_n=-c_n,\, e_n=0, c'_n,\ d'_n,\ e'_n\). \(\square \)
Note that Relations (15) and (16) can also be obtained using the Maple procedure sumdiffrule of the hsum.mpl package (see [16]).
Now let \(m,n\ge 1\). Proceeding as Berg and Vignat [5], we differentiate (11) (for \(\lambda =\mu =\alpha \) and \(\delta =\gamma =\beta \)) to obtain
Using first (15) and then (11), this equation becomes
Since this relation is true for all x, therefore for \(x=0\), we have
This equation is valid if \((ac_n+bc_m)L_0(m,n,a,b)=0\), \(ac_n+bc_m-c_k=0,\, k=1,\ldots ,n+m\), that is, if \(L_0(m,n,a,b)=0\), \(\alpha =0\) and \(a+b=1\) since clearly from the definition (3), \(\beta \ne 0\). Under these conditions, we remain with
Finally dividing the above equation by \(\beta x\) and equating the coefficients of \(\theta _{k}(x;0,\beta )\) yields:
Proposition 5
For \(n,m\ge 1\) and \(k=0,1,\ldots ,n+m-1\), the recurrence equation
with \(L^{(\beta )}_0(m,n,a,1-a)=0\) is satisfied by the linearization coefficients \(L^{(\beta )}_k(n,m,a,1-a)\) of the linearization formula (13).
Remark 6
If the Bessel polynomials \(\theta _n(x;\alpha ,\beta )\) were orthogonal, this method for computing the recurrence relation of the linearization coefficients could be extended to the case \(a+b\ne 1\). In fact, Favard’s theorem [10] states that if they are orthogonal, they satisfy a three-term recurrence relation of the form
that we can use to substitute \(x\theta _k(x;\alpha ,\beta )\) in (17) and equate the coefficients of \(\theta _k(x;\alpha ,\beta )\) to get a mixed recurrence equation for the linearization coefficients.
If the recurrence equation (19) was valid, then to get the coefficients \(A_n, B_n, C_n\), we substitute \(\theta _n(x;\alpha ,\beta )=k_nx^n+k'_nx^{n-1}+k''_nx^{n-2}+\cdots \) in (19) and equate the coefficients of \(x^n, x^{n-1}, x^{n-2}\) to get \(A_n, B_n, C_n\). But it happens that this system has no solution meaning that such three-term recurrence relation doesn’t exist for the family \(\theta _k(x;\alpha ,\beta )\). Zeilberger’s algorithm (see e.g. [16, 21]) deals with sums of the form
and generates a homogeneous linear recurrence equation with polynomial coefficients for \(S_n\). Using Zeilberger’s algorithm implemented in Maple by the sumrecursion procedure of the hsum package [16], we find that the recurrence equation satisfied by \(\theta _n(x;\alpha ,\beta )\) is
This equation can also be obtained by substituting \(x\leftarrow \frac{2}{\beta x}\) in [17, Eq. (9.13.3), p. 245] and multiplying the resulting equation by \(x^{n+1}\).
Remark 7
With the help of a computer algebra system, extensive numerical investigations indicate that for all a and \(\beta \) in (13), the coefficients \(L^{(\beta )}_k(m,n,a,1-a)=0, 0\le k < \min (m,n), m,n\ge 1\).
Theorem 8
Let \(m,n\ge 1\). Then for \(0\le a \le 1\) and \(\beta >0, L^{(\beta )}_k(m,n,a,1-a)\ge 0, k=0,1,\ldots ,n+m\).
The proof of this theorem uses the following result.
Proposition 9
For \(k=0,1,\ldots ,n,\) the coefficients
of the multiplication formula
obtained from (12) by taking \(\alpha =0\) are nonnegative for \(0\le a\le 1\) and \(\beta >0\).
Proof
In fact for \(0\le a\le 1\) and \(\beta >0\),
It remains to prove that
From the formula (see e.g. [22, Eq. (3), p. 388])
we have for \(a_1=-k-1,\, a_2=-n+k,\, a_3=k+2,\, b_1=-2n+k,\, b_2=k+1,\ \omega =1,\, z=a\),
Note that the sum over j is now from 0 to \(k+1\) since \((-k-1)_j=0\) when \(j>k+1\). From the following formula (see e.g. [22, Eq. (82), p. 539])
for \(n=j, b=k+2, c=-n+k, d=-2n+k, e=k+1\), it follows that
This sum contains only two summands and simplifies to
We therefore deduce that
which is clearly nonnegative for \(0\le a\le 1\). \(\square \)
Proof of Theorem 8
Let \(0\le a\le 1\) and \(\beta >0\). The nonnegativity of the coefficients \(L^{(\beta )}_k(m,n,a,1-a)\) follows by induction (on m and n) from the nonnegativity of \(D_k(n,a,\beta )=L^{(\beta )}_k(0,n,a,1-a)\), \(D_k(m,1-a,\beta )=L^{(\beta )}_k(m,0,a,1-a)\) and the recurrence relation (18). \(\square \)
Proposition 10
The linearization coefficients \(L_k^{(\alpha )}(m,n,a,b)\) of the linearization formula (14) are solution of the mixed recurrence equation
Proof
Differentiate the both sides of (14) and multiply the result by \(x^2\) to use the structure relation (16). Rewrite the output with the help of (14) and use the three-term recurrence equation [17, Eq. (9.13.3)]
Equating the coefficients of \(y_k(x;\alpha )\) leads to the result. \(\square \)
Remark 6 may explain why some researchers are interested by the case \(b=1-a\). The special case \(\alpha =\lambda =\mu =0\) and \(\beta =\delta =\gamma =2\) leads to the Bessel polynomials \(q_n(x)\) defined in (9) and Eq. (11) becomes (see [4])
with
For \(b=1-a\), that is, for \(0\le a \le 1\) and \(\beta >0\), since \(L^{(\beta )}_k(m,n,a,1-a)\ge 0\) , we deduce for \(\beta =2\) from the relation \(L_k(m,n,a,1-a) =\frac{2^{n+m}n!m!(2k)!}{(2n)!(2m)!2^kk!} L^{(2)}_k(m,n,a,1-a)\) that \(L_k(m,n,a,1-a)\ge 0\) for \(0\le a \le 1\) and \(L_k(m,n,a,1-a) = 0\) for \(k<\min (m,n)\), which are the results obtained by Berg and Vignat [5]. Equation (20) is therefore a generalization of their linearization formula for all \(a>0\) and \(b>0\) and Theorem 8 a more general result of the nonnegativity of the linearization coefficients for all \(\beta >0\).
Berg and Vignat [5] wrote that they “were unable to derive the explicit expression of the linearization coefficients \(L_k(m,n,a,1-a)\)” of the polynomial family \((q_n(x))_n\) in (20). Nevertheless, they proved using a recursion formula for \(L_k(m,n,a,1-a)\) (see [5, Lemma 3.6], [3, Eq. (5)])
for \(k=0,1,\ldots ,m+n-1\) with \(L_0(m,n,a,1-a)=0\), \(L_{n+m}(m,n,a,1-a)=a^n(1-a)^m\), that these coefficients are nonnegative for \(0\le a\le 1\) and \(L_k(m,n,a,1-a) = 0\) if \(k<\min (m,n)\). Using this nonnegativity, they deduced that the distribution of a finite convex combination of independent Student t-random variables with arbitrary odd degrees of freedom has a density which is a finite convex combination of certain Student t-densities with odd degrees of freedom. Atia and Zeng [3] were the first to derive a double sum formula for \(L_k(m,n,a,1-a)\) that they simplified to the single sum
Note that the explicit expression of \(L_k(m,n,a,b)\) in (20) generalizes the result by Atia and Zeng and in the case \(b=1-a\), it reduces to
By direct computation, we get as in [3, p. 216]
We make a conjecture after further computation that (23) is equivalent to (22). In fact, from the relation \( L_k(m,n,a,1-a) =\frac{2^{n+m}n!m!(2k)!}{(2n)!(2m)!2^kk!} L^{(2)}_k(m,n,a,1-a)\) and the recurrence relation (18), it follows that \(L_k(m,n,a,1-a)\) is solution of the recurrence relation (21). Since (22) and (23) are solution of the same recurrence equation (21) with the same initial conditions, it follows that they are equivalent.
One question arise at this point: Is-it also possible to reduce the linearization coefficients \(L^{(\beta )}_k(m,n,a,1-a)\) given in (13) as double sum into a single sum? In this direction, we get
with \(L_{n+m}(m,n,a,1-a)\), \(L_{n+m-1}(m,n,a,1-a)\), \(L_{n+m-2}(m,n,a,1-a)\) given, respectively, by (24)–(26). It follows from (11) that
with
The latter expression and Eq. (22) lead to
from which we deduce for \(b=1-a\)
which is the single sum expression of \(L^{(\beta )}_{k}(m,n,a,1-a)\).
In the recent manuscript [4], Benabdallah and Atia derived the linearization formula (20) and wrote the linearization coefficients \(L_k(m,n,a,b)\) as a triple sum from which they deduce the following positivity result.
Corollary 11
(see [4]) For \(0\le k\le n+m\) and \(a+b\ne 0\), the linearization coefficients \(L_k(m,n,a,b)\) of (20) are positive if and only if \(0\le a\le a+b\le 1\).
From the above corollary and Relation (28), we deduce that
Corollary 12
For \(0\le k\le n+m\), \(a+b\ne 0\) and \(\beta >0\), the linearization coefficients \(L^{(\beta )}_{k}(m,n,a,b)\) of the linearization formula (27) are positive if and only if \(0\le a\le a+b\le 1\).
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The author is grateful for the useful comments from the referee which improve considerably this manuscript.
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This work was supported by the Institute of Mathematics of the University of Kassel to whom I am very grateful.
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Tcheutia, D.D. Nonnegative linearization coefficients of the generalized Bessel polynomials. Ramanujan J 48, 217–231 (2019). https://doi.org/10.1007/s11139-018-0006-y
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DOI: https://doi.org/10.1007/s11139-018-0006-y