Explicit construction

For real α, β and n = 0, 1, 2, ..., a function R^(α,β)
_n(x) can be defined by the Rodrigues formula in Equation (4) as

${\displaystyle R_{n}^{(\alpha ,\beta )}(x)\equiv {\frac {1}{w^{(\alpha ,\beta )}(x)}}{\frac {{\rm {d}}^{n}}{{\rm {d}}x^{n}}}\left(w^{(\alpha ,\beta )}(x)s(x)^{n}\right),}$

(10)

where w^(α,β) is the same weight function as in (2), and s(x) = 1 + x² is the coefficient of the second derivative of the hypergeometric differential equation as in (1).

Note that we have chosen the normalization constants N_n = 1, which is equivalent to making a choice of the coefficient of highest degree in the polynomial, as given by equation (5). It takes the form

${\displaystyle c_{n}={\frac {1}{n!}}\prod _{k=0}^{n-1}{\bigl (}2\beta (n-k)+n(n-1)-k(k-1){\bigr )},\quad n\geq 1.}$

(11)

Also note that the coefficient c_n does not depend on the parameter α, but only on β and, for particular values of β, c_n vanishes (i.e., for all the values

${\displaystyle \beta ={\frac {k(k-1)-n(n-1)}{2(n-k)}}}$

where k = 0, ..., n − 1). This observation poses a problem addressed below.

For later reference, we write explicitly the polynomials of degree 0, 1, and 2,

${\displaystyle {\begin{aligned}R_{0}^{(\alpha ,\beta )}(x)&=1,\\[6pt]R_{1}^{(\alpha ,\beta )}(x)&={\frac {1}{w^{(\alpha ,\beta )}(x)}}\left(w'^{(\alpha ,\beta )}(x)s(x)+s'(x)w^{(\alpha ,\beta )}(x)\right)\\[6pt]&=t^{(\alpha ,\beta )}(x)=2\beta x+\alpha ,\\[6pt]R_{2}^{(\alpha ,\beta )}(x)&={\frac {1}{w^{(\alpha ,\beta )}(x)}}{\frac {\mathrm {d} }{{\mathrm {d} }x}}\left(s^{2}(x)w'^{(\alpha ,\beta )}(x)+2s(x)s'(x)w^{(\alpha ,\beta )}(x)\right)\\&={\frac {1}{w^{(\alpha ,\beta )}(x)}}{\frac {\mathrm {d} }{{\mathrm {d} }x}}\left(s(x)w^{(\alpha ,\beta )}(x)\left(t^{(\alpha ,\beta )}(x)+s'(x)\right)\right)\\[6pt]&=\left(2x+t^{(\alpha ,\beta )}(x)\right)t^{(\alpha ,\beta )}(x)+\left(2+t'^{(\alpha ,\beta )}(x)\right)s(x)\\[6pt]&=(2\beta +1)(2\beta +2)x^{2}+2(2\beta +1)\alpha x+\left(2\beta +\alpha ^{2}+2\right),\end{aligned}}}$

which derive from the Rodrigues formula (10) in conjunction with Pearson's ODE (3).

Orthogonality

The two polynomials, R^(α,β)
_m(x) and R^(α,β)
_n(x) with m ≠ n, are orthogonal,^[3]

${\displaystyle \int _{-\infty }^{+\infty }w^{(\alpha ,\beta )}(x)R_{m}^{(\alpha ,\beta )}(x)R_{n}^{(\alpha ,\beta )}(x)=0}$

(12)

if and only if,

${\displaystyle m+n<1-2\beta .}$

(13)

In other words, for arbitrary parameters, only a finite number of Romanovski polynomials are orthogonal. This property is referred to as finite orthogonality. However, for some special cases in which the parameters depend in a particular way on the polynomial degree infinite orthogonality can be achieved.

This is the case of a version of equation (1) that has been independently encountered anew within the context of the exact solubility of the quantum mechanical problem of the trigonometric Rosen–Morse potential and reported in Compean & Kirchbach (2006).^[17] There, the polynomial parameters α and β are no longer arbitrary but are expressed in terms of the potential parameters, a and b, and the degree n of the polynomial according to the relations,

${\displaystyle {\begin{aligned}\alpha \to \alpha _{n}={\frac {2b}{n+1+a}},\quad \beta \to \beta _{n}=-(a+n+1)+1,\quad n=0,1,2,\ldots ,\infty .\end{aligned}}}$

(14)

Correspondingly, λ_n emerges as λ_n = −n(2a + n − 1), while the weight function takes the shape

${\displaystyle \left(1+x^{2}\right)^{-(a+n+1)}\exp \left(-{\frac {2b}{n+a+1}}\operatorname {arccot} x\right).}$

Finally, the one-dimensional variable, x, in Compean & Kirchbach (2006)^[17] has been taken as

${\displaystyle x=\cot \left({\frac {r}{d}}\right),}$

where r is the radial distance, while ${\displaystyle d}$ is an appropriate length parameter. In Compean & Kirchbach^[17] it has been shown that the family of Romanovski polynomials corresponding to the infinite sequence of parameter pairs,

${\displaystyle \left(\alpha _{1},\beta _{1}\right),\left(\alpha _{2}\beta _{2}\right),\ldots ,\left(\alpha _{n}\beta _{n}\right),\ldots ,\quad n\longrightarrow \infty ,}$

(15)

is orthogonal.

Generating function

In Weber (2007)^[18] polynomials Q^{(α_n, β_n + n)}
_ν(x), with β_n + n = −a, and complementary to R^{(α_n, β_n)}
_n(x) have been studied, generated in the following way:

${\displaystyle Q_{\nu }^{\left(\alpha _{n},\beta _{n}+n\right)}(x)={\frac {1}{w^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}}}{\frac {{\mathrm {d} }^{\nu }}{{\mathrm {d} }x^{\nu }}}w^{\left(\alpha _{n},\beta _{n}\right)}(x)\left(1+x^{2}\right)^{n}.}$

(16)

In taking into account the relation,

${\displaystyle w^{\left(\alpha _{n},\beta _{n}\right)}(x)\left(1+x^{2}\right)^{\delta }=w^{\left(\alpha _{n},\beta _{n}+\delta \right)}(x),}$

(17)

Equation (16) becomes equivalent to

${\displaystyle {\begin{aligned}Q_{\nu }^{\left(\alpha _{n},\beta _{n}+n\right)}(x)&={\frac {1}{w^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}}}{\frac {{\mathrm {d} }^{\nu }}{{\mathrm {d} }x^{\nu }}}w^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}(x)\left(1+x^{2}\right)^{\nu }\\[4pt]&=R_{\nu }^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}(x),\end{aligned}}}$

(18)

and thus links the complementary to the principal Romanovski polynomials.

The main attraction of the complementary polynomials is that their generating function can be calculated in closed form.^[19] Such a generating function, written for the Romanovski polynomials based on Equation (18) with the parameters in (14) and therefore referring to infinite orthogonality, has been introduced as

${\displaystyle G^{\left(\alpha _{n},\beta _{n}\right)}(x,y)=\sum _{\nu =0}^{\infty }R_{\nu }^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}(x){\frac {y^{\nu }}{\nu$ !}}.}

(19)

The notational differences between Weber^[18] and those used here are summarized as follows:

• G^{(α_n, β_n)}(x,y) here versus Q(x,y;α,−a) there, α there in place of α_n here,
• a = −β_n − n, and
• Q^(α,−a)
  _ν(x) in Equation (15) in Weber^[18] corresponding to R^{(α_n, β_n + n − ν)}
  _ν(x) here.

The generating function under discussion obtained in Weber^[18] now reads:

${\displaystyle G^{(\alpha _{n},\beta _{n})}(x,y)=\left(1+x^{2}\right)^{-\beta _{n}-n+1}\exp \left(\alpha _{n}\operatorname {arccot} x\right)\left(1+\left(x+y\left(1+x^{2}\right)\right)^{2}\right)^{-\left(-\beta _{n}-n+1\right)}\exp \left(-\alpha _{n}\operatorname {arccot} \left(x+y\left(1+x^{2}\right)\right)\right.}$

(20)