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API REFERENCE/omnibias-qcalculus

omnibias.qcalculus.umbral

0.1.0a2AlphaPython >=3.10Apache-2.0

q-numbers, Jackson derivatives and integrals, q-series, and the ordinary-calculus limit as q approaches one.

import omnibias.qcalculus.umbral
Module contract & mathematical conventions

omnibias.qcalculus.umbral -- the q-umbral / q-Sheffer calculus surface (public namespace).

The quantum twin of omnibias.difference.umbral, gathered into one public submodule:

  • **q-Stirling numbers & factorial bases** -- q_stirling_second, q_stirling_first_signed, their rows, and q_falling_factorial_coeffs / q_rising_factorial_coeffs.
  • **q-transforms** -- q_binomial_transform / q_inverse_binomial_transform, q_monomial_to_falling / q_falling_to_monomial, q_newton_forward_coeffs / q_newton_forward_value.
  • **q-Sheffer sequences** -- QShefferClass / q_sheffer_classify, q_sheffer_sequence, q_associated_sequence, q_appell_sequence, q_umbral_composition.
  • **q-umbral operators** -- q_delta_operator_apply (Q = f(D_q)), q_pincherle_derivative.

Everything is exact ~fractions.Fraction arithmetic (**closed-form**), and carries the exact q-Sheffer recurrence Q s_n = [n]_q s_{n-1}. Every object reduces to its classical omnibias.difference.umbral counterpart as **q -> 1**. That is a **distinct** limit from the delta -> 0 founding bias-collapse of omnibias.difference and the beta -> inf **temperature collapse** feasibility penalty of the optimization packages -- same "collapse" word, different limits, never conflated. Every symbol here is also re-exported flat from omnibias.qcalculus.

Shared formulas do not imply universal floating-point identity. Verify dtype, shape, derivative-order support, and the stated numerical assumptions for the path you use.

Module source ↗Source revision 9a2d73df0573Running reference examples →

Imports & aliases​

These names share a definition with another public module. Follow a name for its full contract, parameters, and examples.

Import from this moduleCanonical reference
QShefferClassomnibias.qcalculus.QShefferClass
q_appell_sequenceomnibias.qcalculus.q_appell_sequence
q_associated_sequenceomnibias.qcalculus.q_associated_sequence
q_binomial_transformomnibias.qcalculus.q_binomial_transform
q_delta_operator_applyomnibias.qcalculus.q_delta_operator_apply
q_falling_factorial_coeffsomnibias.qcalculus.q_falling_factorial_coeffs
q_falling_to_monomialomnibias.qcalculus.q_falling_to_monomial
q_inverse_binomial_transformomnibias.qcalculus.q_inverse_binomial_transform
q_monomial_to_fallingomnibias.qcalculus.q_monomial_to_falling
q_newton_forward_coeffsomnibias.qcalculus.q_newton_forward_coeffs
q_newton_forward_valueomnibias.qcalculus.q_newton_forward_value
q_pincherle_derivativeomnibias.qcalculus.q_pincherle_derivative
q_rising_factorial_coeffsomnibias.qcalculus.q_rising_factorial_coeffs
q_sheffer_classifyomnibias.qcalculus.q_sheffer_classify
q_sheffer_sequenceomnibias.qcalculus.q_sheffer_sequence
q_stirling_first_signedomnibias.qcalculus.q_stirling_first_signed
q_stirling_first_signed_rowomnibias.qcalculus.q_stirling_first_signed_row
q_stirling_first_unsignedomnibias.qcalculus.q_stirling_first_unsigned
q_stirling_secondomnibias.qcalculus.q_stirling_second
q_stirling_second_rowomnibias.qcalculus.q_stirling_second_row
q_umbral_compositionomnibias.qcalculus.q_umbral_composition
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