omnibias.difference.umbral
Finite-difference stencils, confluence limits, polynomial sequences, and scoped derivative-extraction certificates.
import omnibias.difference.umbralModule contract & mathematical conventions
omnibias.difference.umbral -- the umbral / Sheffer calculus surface (public namespace).
The discrete counterpart of the derivative tower, gathered into one public submodule. It
pairs the umbral *transforms* (from omnibias.difference._core.umbral) with the
*generative* operator layer (from omnibias.difference._core.sheffer):
- **Finite differences / interpolation** --
forward_difference,newton_forward_coeffs,newton_forward_value,binomial_coefficient. - **Transforms** --
binomial_transform/inverse_binomial_transform,monomial_to_falling/falling_to_monomial(the Stirling transforms),connection_constants. - **Formal power series** --
compose_series,compositional_inverse,series_reciprocal. - **Sheffer sequences** --
ShefferClass/sheffer_classify,sheffer_sequence,associated_sequence,appell_sequence,umbral_composition. - **Umbral operators** --
shift_polynomial(E^a),delta_operator_apply(Q = f(D)),pincherle_derivative(Q' = f'(D)),umbral_functional(the linear functional<L | x^k> = mu_k). - **Riordan group** --
riordan_array,riordan_product,riordan_inverse.
Everything is exact ~fractions.Fraction arithmetic (**closed-form**). This is the
founding delta -> 0 derivative register -- the biases coalesce and the finite difference
*becomes* the derivative. It is **not** the beta -> inf feasibility penalty of the
optimization packages, nor the q -> 1 deformation of
omnibias.qcalculus.umbral (its quantum twin); same "collapse" word, different limits,
never conflated. Every symbol here is also re-exported flat from omnibias.difference
for backward compatibility.
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.
Imports & aliases
These names share a definition with another public module. Follow a name for its full contract, parameters, and examples.
| Import from this module | Canonical reference |
|---|---|
ShefferClass | omnibias.difference.ShefferClass |
appell_sequence | omnibias.difference.appell_sequence |
associated_sequence | omnibias.difference.associated_sequence |
binomial_coefficient | omnibias.difference.binomial_coefficient |
binomial_transform | omnibias.difference.binomial_transform |
compose_series | omnibias.difference.compose_series |
compositional_inverse | omnibias.difference.compositional_inverse |
connection_constants | omnibias.difference.connection_constants |
delta_operator_apply | omnibias.difference.delta_operator_apply |
falling_to_monomial | omnibias.difference.falling_to_monomial |
forward_difference | omnibias.difference.forward_difference |
inverse_binomial_transform | omnibias.difference.inverse_binomial_transform |
monomial_to_falling | omnibias.difference.monomial_to_falling |
newton_forward_coeffs | omnibias.difference.newton_forward_coeffs |
newton_forward_value | omnibias.difference.newton_forward_value |
pincherle_derivative | omnibias.difference.pincherle_derivative |
riordan_array | omnibias.difference.riordan_array |
riordan_inverse | omnibias.difference.riordan_inverse |
riordan_product | omnibias.difference.riordan_product |
series_reciprocal | omnibias.difference.series_reciprocal |
sheffer_classify | omnibias.difference.sheffer_classify |
sheffer_sequence | omnibias.difference.sheffer_sequence |
shift_polynomial | omnibias.difference.shift_polynomial |
umbral_composition | omnibias.difference.umbral_composition |
umbral_functional | omnibias.difference.umbral_functional |