1 Introduction
The properties of generalised Bessel polynomials have been studied extensively [Reference Grosswald6, Reference Srivastava14] and they have applications in solutions to the wave equation [Reference Burchnall2, Reference Krall and Frink8] and signal processing filters.
The contribution of this work is to give inverse connection formulae. They provide coefficients for the mappings from the monomials to either the generalised Bessel polynomials or the reverse generalised Bessel polynomials.
The connection formulae enable us to derive change of basis matrices directly from any sequence of polynomials that spans
$(1, x, \ldots , x^n)$
to a basis that spans the same vector space and includes polynomials from either family of generalised Bessel polynomials.
Suppose that
$(f_0(x), f_1(x), \ldots , f_n(x))$
is a sequence of polynomials over
$\mathbb {R}[x]$
that spans the same vector space as
$(1, x, \ldots , x^n)$
and where
$f_i(x)$
has degree i in x for
$0 \leq i \leq n$
. The inverse connection formula we consider is to find a function
$\alpha _1: \{0, 1, \ldots , n\}^2 \rightarrow \mathbb {R}$
such that
$$ \begin{align*} x^k = \sum_{v=0}^k \alpha_1(k, i)f_{k-i}(x), \end{align*} $$
where
$0 \leq k \leq n$
.
Suppose we have a sequence of polynomials
$(h_0(x), h_1(x), \ldots , h_n(x))$
that spans
$(1, x, \ldots , x^n)$
where
$$ \begin{align*} h_i(x) = \sum_{v=0}^i \alpha_2(i, v) x^{i - v} \end{align*} $$
and
$\alpha _2: \{0, 1, \ldots , n\}^2 \rightarrow \mathbb {R}$
. The basis
$(h_0(x), h_1(x), \ldots , h_n(x))$
could include classical orthogonal polynomials such as Chebyshev polynomials of the first kind or nonorthogonal polynomials such as Bernstein polynomials. We select the Laguerre polynomials
$(L_0(x), L_1(x), \ldots , L_n(x))$
as a running example, so that
$$ \begin{align} \alpha_2(n, k) = \frac{(-1)^{n-k}}{(n-k)!} {{n } \choose k}. \end{align} $$
This equation follows from [Reference Koornwinder, Wong, Koekoek and Swarttouw7, (18.18.18)]. We have
$$ \begin{align} \alpha(n,k) = \sum_{v=0}^k \alpha_1(n-v,k-v) \; \alpha_2(n,v). \end{align} $$
The element
$m_{i,j}$
of the change of basis matrix from
$(h_0(x), h_1(x), \ldots , h_n(x))$
to
$(f_0(x), f_1(x), \ldots , f_n(x))$
is given by
$$ \begin{align*} m_{i,j} = \begin{cases} \alpha(\,j, j-i)& \mbox{if}\ j \geq i,\\ 0 & \mbox{if}\ j < i,\\ \end{cases} \end{align*} $$
where
$0 \leq i, j \leq n$
.
The right-hand side of (1.2) is an expression for the element in the row with index
$n-k$
of the last column of the product of two upper triangular change of basis matrices: that from
$(1, x, \ldots , x^n)$
to
$(f_0(x), f_1(x), \ldots , f_n(x))$
with that from
$(h_0(x), h_1(x), \ldots , h_n(x))$
to
$(1, x, \ldots , x^n)$
.
The elements can be found more directly because (1.2) separates finding
$\alpha $
into two sub-problems both of which can be solved independently, that is, finding solutions for
$\alpha _1$
and
$\alpha _2$
. We now solve the sub-problem for
$\alpha _1$
for both kinds of generalised Bessel polynomials.
2 Generalised Bessel polynomials
The generalised Bessel polynomials can be defined by
$$ \begin{align} y_n(x; \alpha, \beta) = \sum_{l=0}^n {n \choose l} (n + \alpha - 1)_l \bigg(\frac{x}{\beta}\bigg)^l, \end{align} $$
where
$(x)_l$
is the Pochhammer symbol or rising factorial. This is derived from Krall and Frink [Reference Krall and Frink8, (34)].
The connection formula for the mapping to the monomials is therefore
$$ \begin{align} \alpha(n, k, \alpha, \beta) = {n \choose k} \frac{(n + \alpha - 1)_{n-k}}{{\beta}^{n-k}}. \end{align} $$
For the inverse connection formula for the mapping from monomials, we use the following equation from Doha and Ahmed [Reference Doha and Ahmed5, (8)] as the starting point for an Ansatz. It is equivalent to that of Sánchez-Ruiz and Dehesa [Reference Sánchez-Ruiz and Dehesa13, (2.32)]:
$$ \begin{align*} \pi_{ni} = {n \choose i} \frac{(-1)^{n-i}2^n (2i + \alpha + 1) \Gamma(i + \alpha + 1)}{\Gamma(n + i + \alpha + 2)}. \end{align*} $$
This is the inverse connection formula for the term
$y_i(x; \alpha +2, 2)$
, where
$0 \leq i \leq n$
. To find the Ansatz, we replace i by
$n-k$
to be consistent with our notation,
$\alpha $
by
$\alpha - 2$
, the ratio of gamma functions by a Pochhammer symbol and the constant
$2$
by
$\beta $
. This gives the equation
$$ \begin{align} \alpha(n, k, \alpha, \beta) = {n \choose k} \frac{(-1)^{k}\beta^n (2(n-k) + \alpha -1)}{(n - k + \alpha - 1)_{n+1}}. \end{align} $$
Theorem 2.1. Equation (2.3) is the connection formula for the mapping from monomials to generalised Bessel polynomials.
Proof. The proof is by mathematical induction. It uses back substitution to find the connection formula for the last column of the inverse matrix of the change of basis matrix from generalised Bessel polynomials to the monomials. For clarity, we call the function in (2.2)
$\alpha _0$
.
In the base case,
$$ \begin{align*} \alpha_1(n, 0, \alpha, \beta) = \frac1{{n \choose 0} \tfrac{(n + \alpha -1)_n}{\beta^n}}, \end{align*} $$
from (2.2). This equals the
$\alpha (n, 0, \alpha , \beta )$
from (2.3) when
$k = 0$
, as required.
The induction hypothesis is that (2.3) holds for all
$k: 0\leq k < m \leq n$
. From this induction hypothesis and back substitution,
$$ \begin{align} \alpha_1(n, m, \alpha, \beta) = \frac{\sum_{k = 1}^{m} \alpha_0(n-m+ k, k, \alpha, \beta)\alpha(n, m - k, \alpha, \beta)}{-\alpha_0(n-m, 0, \alpha, \beta)}. \end{align} $$
We find that
$\alpha _1(n, m, \alpha , \beta ) = \alpha (n, m, \alpha , \beta )$
by applying Gosper’s algorithm from the ‘fastZeil’ package [Reference Paule, Schorn and Riese12] to the right-hand side of (2.4) and simplifying. The result holds for the base case and from the induction hypothesis for
$k=m$
. By the principle of mathematical induction, it holds for all
$k: 0\leq k \leq n$
.
As an example,
$$ \begin{align*} x^2 &= \sum_{k=0}^2 \alpha(2, k, \alpha, \beta) y_{2-k}(x; \alpha, \beta) \\ &= \frac{\beta^2}{\alpha (1+\alpha)}- \frac{2 \beta^2 (1+\tfrac{\alpha x}{\beta})}{\alpha (2+\alpha)}+ \frac{\beta^2 (1+\tfrac{2 (1+\alpha) x}{\beta}+ \frac{(1+\alpha) (2+\alpha) x^2}{\beta^2})}{(1+\alpha) (2+\alpha)}. \end{align*} $$
3 Reverse generalised Bessel polynomials
The reverse generalised polynomials
$\theta _n(x; \alpha , \beta )$
are the reciprocal polynomials of the generalised Bessel polynomials, that is,
$$ \begin{align} \theta_n(x; \alpha, \beta) = x^n y_n(x^{-1}; \alpha, \beta), \end{align} $$
where
$n \geq 0$
. Burchnall [Reference Burchnall2, Section 5] discussed
$\theta _n(x; \alpha , \beta )$
(see also Grosswald [Reference Grosswald6, Ch. 1] for historical details and the overview of Srivastava [Reference Srivastava14, (15)]). It follows from (2.1) and (3.1) that the reverse polynomials can be defined by
$$ \begin{align*} \theta_n(x; \alpha, \beta) = \sum_{k=0}^n {n \choose k} \frac{x^k}{\beta^{n-k}} (n + \alpha -1)_{n-k}, \end{align*} $$
so that the connection formula to the monomials is
$$ \begin{align} \alpha(n, k, \alpha, \beta) = {n \choose k} \frac{(n + \alpha -1)_{k}}{\beta^{k}}. \end{align} $$
We want to find the connection formula for the inverse mapping, that is, from the monomials to reverse generalised Bessel polynomials. To do this, we use back substitution to find the connection formula for the last column of the inverse matrix of the change of basis matrix from reverse generalised Bessel polynomials to the monomials. We call the function in (3.2)
$\alpha _0$
, to distinguish it from the inverse connection formula,
$\alpha _1$
.
First, we find an Ansatz for the inverse connection formula by generalising successive connection formulae found through back substitution. In the base case, we have
$\alpha _1(n, 0, \alpha , \beta ) = 1/{\alpha (n, 0, \alpha , \beta )}$
, from (3.2), so that
$\alpha _1(n, 0, \alpha , \beta ) = 1$
. Equation (3.3) follows from back substitution:
$$ \begin{align} \alpha_1(n, m, \alpha, \beta) &= \frac{\sum_{k = 1}^{m} \alpha_0(n-m+ k, k, \alpha, \beta)\alpha(n, m - k, \alpha, \beta)}{-\alpha_0(n-m, 0, \alpha, \beta)} \nonumber\\ &= -\sum_{k = 1}^{m} \alpha_0(n-m+ k, k, \alpha, \beta)\alpha(n, m - k, \alpha, \beta). \end{align} $$
By successively applying (3.3), we find the following sequence of specific connection formulae:
$$ \begin{align*} \alpha_1(n, 1, \alpha, \beta) &= -\frac{{n \choose 1}(n+\alpha-1)}{\beta},\\ \alpha_1(n, 2, \alpha, \beta) &= \frac{{n \choose 2}(n + \alpha-1)(n + \alpha -4)}{\beta^2},\\ \alpha_1(n, 3, \alpha, \beta) &= -\frac{{n \choose 3}(n + \alpha-1)(n +\alpha-6)(n + \alpha - 5)}{\beta^3}, \\ \alpha_1(n, 4, \alpha, \beta) &= \frac{{n \choose 4}(n + \alpha-1)(n + \alpha-8)(n + \alpha-7)(n + \alpha-6)}{\beta^4}. \end{align*} $$
We use the following generalisation of the sequence from
$\alpha _1(n, 0, \alpha , \beta )$
to
$\alpha _1(n, 4, \alpha , \beta )$
to give the Ansatz:
$$ \begin{align} \alpha_1(n, k, \alpha, \beta) = (-1)^k \frac{{n \choose k}}{\beta^k} (n + \alpha -1) (n + \alpha - 2k)_{k-1}. \end{align} $$
Equation (3.3) cannot be simplified by applying Gosper’s algorithm to its right-hand side. It is an mth-order recurrence and it is not convenient to use in a proof by mathematical induction. We can apply Zeilberger’s algorithm from the ‘fastZeil’ package [Reference Paule, Schorn and Riese12] to (3.3) to obtain a second-order recurrence for the inverse connection formula. The ‘fastZeil’ package generates the following recurrence with the condition that
$k> 1$
:
$$ \begin{align} &k(k+1)(n-k) \mathrm{SUM}[k] + \beta (k+1)(n - k + \alpha -2) \mathrm{SUM}[k+1] \nonumber\\ &\quad= (k-n)(n - 3k + \alpha -2)(n - k + \alpha -1) \alpha_1(n, k, \alpha, \beta). \end{align} $$
Theorem 3.1. Equation (3.4) is the connection formula for the mapping from monomials to the reverse generalised Bessel polynomials.
Proof. The proof is by mathematical induction on k. The case for
$k=0$
was verified above. The induction hypothesis is that the result holds for all
$k: 0 \leq k <m \leq n$
.
The condition that
$k> 1$
for the recurrence is too strict. We can check that (3.5) also holds when
$k=0$
and
$k=1$
. Suppose that
$k> 0$
, apply (3.5), substitute
$\alpha _1(n, k, \alpha , \beta )$
for
$\mathrm {SUM}[k]$
and solve it for
$\mathrm {SUM}[k+1]$
. After simplifications, we find that
$\mathrm {SUM}[k+1] = \alpha _1(n, k+1, \alpha , \beta )$
, as required. The result holds for the base cases, and from the induction hypothesis for
$k=m$
. By the principle of mathematical induction, it holds for all
$k: 0\leq k \leq n$
.
As an example,
$$ \begin{align*} x^2 &= \sum_{k=0}^2 \alpha_1(2, k, \alpha, \beta) \theta_{2-k}(x; \alpha, \beta) \\ &= \bigg(\frac{(1+\alpha) (2+\alpha)}{\beta^2}+\frac{2 (1+\alpha) x}{\beta}+x^2\bigg) - \frac{2 (1+\alpha) (\tfrac{\alpha}{\beta}+x)}{\beta} + \frac{(-2+\alpha) (1+\alpha)}{\beta^2}. \end{align*} $$
4 Related work
The Bessel polynomials,
$y_n(x) = y_n(x; 2, 2)$
, were named in 1949 by Krall and Frink [Reference Krall and Frink8], and had previously appeared (see, for example, Grosswald [Reference Grosswald6]). They are a sequence of polynomials that are orthogonal on the unit circle of the complex plane. The reverse Bessel polynomials were identified by Burchnall and Chaundry [Reference Burchnall and Chaundy3, page 478] (see also Burchnall [Reference Burchnall2, (8)]).
The Bessel polynomials gain significance as one of the four families of classical orthogonal polynomials [Reference Castillo and Petronilho4, Reference Maroni10]. The other families are the Hermite, Jacobi and Laguerre polynomials (see Koornwinder et al. [Reference Koornwinder, Wong, Koekoek and Swarttouw7, Section 18.3]). Castillo and Petronilho [Reference Castillo and Petronilho4, Section 3.2] formalised how the classical orthogonal polynomials comprise these four families only. Maroni [Reference Maroni10, Sections 2 and 6] discussed characterisations of the classical orthogonal polynomials.
The literature on the topic of change of basis between classical orthogonal polynomials is extensive [Reference Maroni and da Rocha11]. There are various other techniques for finding the connection coefficients that are the elements of change of basis matrices, such as matrix methods [Reference Bella and Reis1] and recurrences [Reference Lewanowicz9, Reference Maroni and da Rocha11].
We showed that the equation from Sánchez-Ruiz and Dehesa [Reference Sánchez-Ruiz and Dehesa13, (2.32)] is a special case of the inverse connection formula, (2.3), for the generalised Bessel polynomial. We are not aware of other similar equations in the literature for the Bessel polynomials.
5 Two examples
We give an example of the application of (1.2), where
$\alpha _1$
is the inverse connection formula for the generalised Bessel polynomial of (2.3) and
$\alpha _2$
is the formula for the Laguerre polynomial of (1.1).
On substitution into (1.2),
$$ \begin{align*} \alpha(n,k) &= \sum_{v=0}^k {n-v \choose k-v} \frac{(-1)^{k-v}\beta^{n-v} (2(n-k) + \alpha -1)}{(n - k + \alpha - 1)_{n-v+1}} \frac{(-1)^{n-v}}{(n-v)!} {{n } \choose v} \nonumber\\ &= (-1)^{n+k} \frac{(2(n-k) + \alpha -1)}{(n-k)!} \sum_{v=0}^k {{n } \choose v}\frac{\beta^{n-v} }{(k-v)!(n - k + \alpha - 1)_{n-v+1}}. \end{align*} $$
When
$n=4$
,
$\alpha = 3$
and
$\beta = -\frac 32$
, we obtain the change of basis matrix:
$$ \begin{align*} \begin{bmatrix} 1 & \frac{1}{2} & \frac{3}{32} & -\frac{73}{320} & -\frac{2429}{5120} \\ \\ 0 & \frac{1}{2} & \frac{17}{20} & \phantom{-}\frac{171}{160} & \frac{2629}{2240} \\ \\ 0 & 0 & \frac{9}{160} & \phantom{-}\frac{351}{2240} & \frac{20817}{71680} \\ \\ 0 & 0 & 0 & \phantom{-}\frac{3}{1120} & \frac{23}{2240} \\ \\ 0 & 0 & 0 & \phantom{-}0 & \frac{1}{14336} \\ \end{bmatrix}. \end{align*} $$
From the fourth column,
$$ \begin{align*} L_3(x) &= \sum_{v=0}^3 \alpha(3, v) y_{3-v}\bigg(x; 3, -\frac32\bigg)\\ &= \frac{3}{1120} y_{3}\bigg(x; 3, -\frac32\bigg) + \frac{351}{2240}y_{2}\bigg(x; 3, -\frac32\bigg) + \frac{171}{160}y_{1}\bigg(x; 3, -\frac32\bigg) -\frac{73}{320}y_{0}\bigg(x; 3, -\frac32\bigg) \\ &= \frac{3 (-\frac{560 x^3}{9} +40 x^2 -10 x +1)}{1120} +\frac{351 (\frac{80 x^2}{9}-\frac{16 x}{3} +1)}{2240} +\frac{171}{160} (-2 x + 1) -\frac{73}{320}\\ &=\frac{1}{6} (-x^3+9 x^2 -18 x +6), \mbox{ as required.} \end{align*} $$
Similarly, when
$\alpha _1$
is the inverse connection formula for the reverse generalised Bessel polynomial of (3.4) and
$\alpha _2$
is the formula for the Laguerre polynomial of (1.1),
$$ \begin{align*} \alpha(n,k) = \frac{(-1)^{n+k}}{(n-k)!} \sum_{v=0}^k {{n } \choose v}\frac{(n-v + \alpha -1)(n +v + \alpha - 2k)_{k-v-1} }{\beta^{k-v}(k-v)!} \end{align*} $$
and
$$ \begin{align*} L_3(x) & = \sum_{v=0}^3 \alpha(3, v) \theta_{3-v}\bigg(x; 3, -\frac32\bigg)\\ & = -\frac{1}{6} \theta_{3}\bigg(x; 3, -\frac32\bigg) + -\frac{1}{6}\theta_{2}\bigg(x; 3, -\frac32\bigg) + \frac{25}{9}\theta_{1}\bigg(x; 3, -\frac32\bigg) -\frac{7}{3}\theta_{0}\bigg(x; 3, -\frac32\bigg) \\ & = \frac{ (x^3 - 10 x^2 + 40x - \frac{560}{9})}{6} -\frac{ (x^2 - \frac{16 x}{3} +\frac{80}{9})}{6} +\frac{25}{9} (x-2) -\frac{7}{3}. \end{align*} $$
Acknowledgements
The author is grateful to the referee for the review of this paper, and to the ANU College of Engineering, Computing and Cybernetics at the Australian National University for research support.
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