{'fraction_non_alphanumeric': 0.04607957870099473, 'fraction_numerical': 0.013750731421884143, 'mean_word_length': 4.24712202609363, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
One-parameter isospectral special functions

Using a combination of the ladder operators of Pina [Rev. Mex. Fis. 41 (1995) 913] and the parametric operators of Mielnik [J. Math. Phys. 25 (1984) 3387] we introduce second order linear differential equations whose eigenfunctions are isospectral to the special functions of the mathematical physics and illustrate the method with several key examples


Introduction
The use of the factorization method [1] proved to be a powerful tool for extending the class of exactly solvable Sturm-Liouville problems especially in quantum mechanics, where in the form of supersymmetric quantum mechanics led to new potentials, which are isospectral to a given problem [2]. In a paper by Mielnik [3], the usual factorization operators of the one dimensional harmonic oscillator Hamiltonian have been replaced by new operators In order that these new operators factorize the same Hamiltonian (2) bb * = aa * = H + 1 2 the function β(x) should satisfy a Riccati equation of the form Using the solutions of this equation γ + x 0 e −y 2 dy (6) one can introduce a new Hamiltonian H, which is defined by the inverse factorization of (4) with new potential functions and whose eigenfunctions are isospectral to the harmonic oscillator eigenfunctions ψ n .

Factorization of special functions
The great majority of differential equations appearing in Physics can be factorized by means of ladder operators. Therefore, we should be able to apply the procedure described in the previous section to the raising and lowering operators of the important class of Sturm-Liouville problems and get in this way isospectral second order differential equations. To attain this objective we proceed as follows.
One of the possible forms of factorizing a subclass of second order differential operators associated to the special functions of mathematical physics was introduced by Piña [4]. Consider the Sturm-Liouville problem where P , Q and R n are functions of the variable x, and R n depend on the index n. Then, it is possible to construct raising and lowering operators [4] that can factorize Eq.(10) in two ways where the constant K n is the same in both factorizations. Furthermore, the functions a + n , a − n+1 , turn out to be where c n is an integration constant. From Eq.(12), one may consider the constant K n as the eigenvalue corresponding to the eigenfunction ψ n for the operator A − n+1 A + n . Let us now define new operators B + n , B − n+1 by and demand that they can also factorize the Sturm-Liouville operator L n (x) as Then, the following relationship should be fulfilled Eq.(20) is a Riccati type equation, which can be easily solved to get Here, δ(x) is defined by the indefinite integral and γ is an integration constant. x 0 may be chosen as the point where the integrand vanishes.
Similarly to Mielnik's new Hamiltonian H, we now introduce the second order differential operator L n (x) given by That is, the new operator L n differs from L n by the derivative of the solution of the Riccati equation (20), in the same way as Mielnik's Hamiltonian H differs from the harmonic oscillator Hamiltonian H, as can be seen in Eq.(7).
If we now define the functions where ψ n are the eigenfunctions of L n , we can see that Therefore, these ψ n are the eigenfunctions for the new operator L n , and we can write the eigenvalue equation As can be seen from Eq.(21), we have constructed a one paramenter family of operators L n (x; γ) which are "isospectral" to the original Sturm-Liouville operator L n (x), for the allowed values of the parameter γ that produce a non-divergent function b + n (x). As we can see, B + n , B − n+1 are not ladder operators as are A + n , A − n+1 . However, one can easily verify that the third order operators play the role of raising and lowering operators, respectively, for the functions ψ n (x). The use of ladder operators of order higher than two is not easy to find in the literature, but some work in this direction has already been reported [5].
Since B − n+1 , and therefore C − n+1 , is not defined for n = 0, the space defined by the eigenfunctions ψ n lack the element with n = 0, similarly to what happens in SUSY factorization [2].

Examples
Here, we proceed to find the new operators L n (x; γ) and their eigenfunctions, from the factorizations of the special functions of mathematical physics.

Hermite polynomials
The Hermite differential equation has raising and lowering differential operators given by In this case, the δ-integral in Eq. (22) is The integrand in the denominator being positive definite, one should impose the condition |γ| > √ π 2 [3] in order to have a well-defined operator. Now, with the use of Eq.(20) one gets and therefore, the second order differential operator has parametric eigenfunctions given by

Laguerre polynomials
For the Laguerre differential equation the raising and lowering operators are The δ-integral is and hence For x > 0, the integral in the denominator is positive definite, with maximum value ∞ 0 y (α+2n+1) e −y dy = Γ(α + 2n + 2) and, since there is no upper limit for increasing n, we must have γ < 0. The second order differential operator has as parametric eigenfunctions
Hence, the second order differential operator has as parametric eigenfunctions

Chebyshev polynomials
For the Chebyshev differential equation the corresponding raising and lowering operators are In this case, the δ-integral is and, hence The integral in the denominator has maximum value which imposes the condition γ > π. We can thus construct the second order differential operator with the parametric eigenfunctions
Therefore, we find that where the integral in the denominator can be written in terms of the incomplete Beta functions B −1 (u, v − u) and B x (u, v − u). In the particular case v > u, to obtain the range of allowed values for the γ parameter, we see that Therefore, we have to demand |γ| > 2Γ(v−u)Γ(u) . The second order differential operator has the parametric eigenfunctions 3.6 Jacobi polynomials The differential equation defining the Jacobi polynomials has raising and lowering operators given by (1 − x 2 ) d dx + (n + 1)(x + β − α 2n + 2 + α + β ) P αβ n+1 (x) = = 2(n + 1 + α)(n + 1 + β) 2n + 2 + α + β P αβ n (x) .
In this case, we have For the parameter γ, since q > p, we demand that .

Bessel functions
For the Bessel differential equation the raising and lowering operators are The δ-integral is found to be δ = x − n y − n + 1 y dy = ln x −(2n+1) , and therefore with γ = 2nγ ′ ≥ 0.
From here, we can construct a second order differential operator defined by whose parametric eigenfunctions are given by Hence, in the case of Bessel functions the new eigenfunctions J n+1 (x; γ) are not regular at x = 0, except in the case γ = 0, for which L n+1 (x; γ) = L n−1 and J n+1 (x; γ) = J n−1 (x).