Derivative of the Jacobi polynomials with respect to their order and applications to indefinite integration

Technical Summary: Derivative of Jacobi Polynomials with Respect to Order and Applications to Indefinite Integration

Problem Statement
Special functions, particularly those depending on numerical parameters, are fundamental in modeling physical and mathematical processes. While derivatives of special functions with respect to their arguments are standard, derivatives with respect to their parameters (parameter derivatives) are often more difficult to determine. Although parameter derivatives of Jacobi polynomials with respect to parameters $a$ and $b$ are known to yield polynomial forms, the derivative of Jacobi polynomials with respect to their order (traditionally a discrete integer $n$) has not been explicitly formulated in closed form. This paper addresses the gap in determining the first derivative of Jacobi polynomials with respect to their order, treating the order as a continuous real variable.

Methodology
The author employs a three-step analytical approach:

1. Extension of Domain: The discrete order $n$ of the Jacobi polynomial $P_n^{(a,b)}(x)$ is extended to a continuous real variable $\mu$ by expressing the polynomial via the Gaussian hypergeometric function $_2F_1$. This relies on the relation:
   $$P_\mu^{(a,b)}(x) = \frac{\Gamma(a+\mu+1)}{\Gamma(\mu+1)\Gamma(a+1)} {}_2F_1\left(-\mu, a+b+\mu+1, a+1, \frac{1-x}{2}\right)$$
   This extension allows for differentiation with respect to $\mu$ on the positive real axis.
2. Differentiation and Series Expansion: The derivative $\frac{\partial}{\partial \mu} P_\mu^{(a,b)}(x)$ is computed using the product rule. The derivative of the hypergeometric function is evaluated by differentiating its series representation term-by-term. This introduces the digamma function $\psi$.
3. Evaluation at Integer Values: The resulting expression is evaluated at non-negative integer values $\mu = n$. The author carefully handles the limits of the digamma function differences as $\mu \to n$, splitting infinite series into finite sums (for $k \le n$) and infinite tails (for $k > n$). Specific identities for the limits of digamma differences are applied to resolve singularities and simplify the series.

Key Contributions and Results
The primary contribution is the derivation of an explicit, closed-form expression for the derivative of Jacobi polynomials with respect to their order, denoted as $\left[ \frac{\partial}{\partial \mu} P_\mu^{(a,b)}(x) \right]_{\mu=n}$.

• General Form: The resulting derivative is generally non-elementary, containing a generalized hypergeometric function $_3F_2$. The formula combines terms involving the original Jacobi polynomial, finite sums of rational functions involving digamma terms, and the $_3F_2$ term.
• Elementary Special Cases: The paper demonstrates that for specific parameter settings, the non-elementary $_3F_2$ term reduces to elementary functions. Three specific cases for $n=0$ are provided:
  • For $a = -b-3$, the result is a polynomial plus digamma constants.
  • For $a = b = 1$, the result involves logarithmic terms.
  • For $a = b = -1/2$, the result involves inverse trigonometric terms (specifically $\arcsin$).
• Application to Indefinite Integration: The derived order derivative is applied to a recently developed method for resolving indefinite integrals using Wronskians. By identifying the Jacobi differential equation with a specific form solvable by the Wronskian method, the author derives a closed-form representation for the indefinite integral:
  $$\int (1-t)^a (1+t)^b \left[ P_n^{(a,b)}(t) \right]^2 dt$$
  This integral class, which previously lacked a reported closed-form solution in the literature, is resolved by evaluating the Wronskian of the function $u(x, \mu)$ and its derivative with respect to $\mu$. An explicit example for $a=b=1$ and $n=2$ is worked out, confirming the validity of the method.

Significance and Scope
The paper claims significance in two main areas:

1. Mathematical Completeness: It provides the first explicit closed-form expression for the derivative of Jacobi polynomials with respect to their order, a quantity distinct from the known parameter derivatives with respect to $a$ and $b$.
2. Practical Utility: It demonstrates that this derivative is not merely a theoretical construct but a necessary component for resolving a specific class of indefinite integrals involving squared Jacobi polynomials.

The author notes that the result can be extended to other polynomial systems that are special cases of Jacobi polynomials, including Gegenbauer, Legendre, and Chebyshev polynomials (of both first and second kinds). The paper concludes by acknowledging that the hypergeometric representation used has constraints (e.g., $a$ cannot be a negative integer) and suggests alternative representations exist for those specific parameter ranges. The work is presented as a note establishing the derivative and its immediate application to integration, without proposing further experimental or computational extensions beyond the mathematical framework provided.