Generalized Fibonacci Polynomials Toolkit

• Implements the recurrence and Binet machinery for generalized Fibonacci polynomials (GFP) described in the paper Zeros and Orthogonality of generalized Fibonacci polynomials.
• Ships a Wolfram Language package (src/GeneralizedFibonacciPolynomials.wl) that exposes helpers to:
  • construct Fibonacci- and Lucas-type GFP families,
  • generate polynomials via recurrence or Binet formulas,
  • recover generalized Hoggatt (binomial coefficient) expansions without recursion,
  • approximate polynomial zeros,
  • analyse orthogonality weight functions, numerically verify orthogonality via quadrature, and evaluate Karlin–McGregor transition integrals,
  • extract random-walk coefficients when the hypotheses from the reference paper are satisfied, and
  • build truncated stochastic / birth–death models with potential coefficients and basic ergodicity diagnostics.
• Includes a WolframScript test harness (tests/runTests.wls) that validates the main identities against textbook families (e.g. Chebyshev and classical Fibonacci sequences).

Usage

• Ensure wolframscript.exe from Mathematica 14.0 is accessible (default path on this machine: D:\Software\Wolfram Research\Mathematica\14.0\wolframscript.exe).
• From the repository root, run "/mnt/d/Software/Wolfram Research/Mathematica/14.0/wolframscript.exe" -file tests/runTests.wls inside WSL/Arch to execute the automated checks.
• Load the package inside your own notebooks or scripts with Get["src/GeneralizedFibonacciPolynomials.wl"] (or append src to $Path and use Needs["GeneralizedFibonacciPolynomials"]`).
• Construct families through CreateGFPFamily, then call GFPPolynomial, GFPBinet, GFPBinomialExpansion, GFPZeros, GFPOrthogonalityData, GFPOrthogonalityCheck, GFPKnownWeights, GFPKarlinMcGregor, GFPRandomWalkData, or GFPRandomWalkModel as needed.

Repository Layout

• src/GeneralizedFibonacciPolynomials.wl — Wolfram Language package implementing the GFP toolkit.
• tests/runTests.wls — WolframScript test runner.
• FORMULAS.md — TeX-ready reference of the identities encoded in the implementation.
• TEST_SUMMARY.md — Execution report for the automated test suite.
• reference_paper/ — Original arXiv source (ignored by version control per requirements).

Future Extensions

• Automate invariant-measure evaluation by approximating $\sum_i \pi_i$ for both discrete and continuous birth–death models so users can quickly diagnose ergodicity from the potential coefficients.
• Explore duality mappings between Markov processes and GFP families as highlighted in the concluding discussion, packaging reusable duality function builders.