In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.[1]
^See Zeilberger 1990 and Kauers & Paule 2011.
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In mathematics, and more specifically in analysis, a holonomicfunction is a smooth function of several variables that is a solution of a system of linear...
and time t {\displaystyle t\,\!} Holonomic module in the theory of D-modules Holonomicfunction, a smooth function that is a solution of a linear homogeneous...
called holonomicfunctions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special...
have a holonomic generating function are equivalent. Holonomicfunctions are closed under the Hadamard product operation ⊙ on generating functions. The...
process function X may be either holonomic or non-holonomic. For a holonomic process function, an auxiliary state function (or integrating factor) λ may...
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following...
Holonomic brain theory is a branch of neuroscience investigating the idea that human consciousness is formed by quantum effects in or between brain cells...
elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation (see holonomicfunction). Solving a recurrence...
special functions such as Airy function, error function, Bessel functions and all hypergeometric functions. A fundamental property of holonomicfunctions is...
linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomicfunction). As, in general, the solutions...
In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold M is a set of basis vector fields {e1, ....
conservative potential function as can, for example, the inverse square law of the gravitational force. This latter is an example of a holonomic system: path integrals...
special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomicfunction). When...
A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a continuous function g(x),...
their derivatives. Constraints of this type are known as non-holonomic. First-order non-holonomic constraints have the form g ( q , q ˙ , t ) = 0 , {\displaystyle...
de Bruijn prize van der Hoeven, Joris (1999). "Fast evaluation of holonomicfunctions". Theoretical Computer Science. 210: 199–215. doi:10.1016/S0304-3975(98)00102-9...
For most holonomic sequences, there is no explicit formula for expressing a n {\displaystyle a_{n}} as a function of n. Nevertheless, holonomic sequences...
macroscopic ordering often associated with higher cognitive functions in QBD. Karl Pribram's holonomic brain theory (quantum holography) invoked quantum mechanics...
Primary constraint, secondary constraint, etc. in Hamiltonian mechanics Holonomic constraints, also called integrable constraints, (depending on time and...
About the second part: "It is remarkably hard to find the etymology of holonomic (or holonomy) on the web. I found the following (thanks to John Conway...
Pribram, Bohm was involved in the early development of the holonomic model of the functioning of the brain, a model for human cognition that is drastically...
principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral...
known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He was a professor at the Research Institute for Mathematical...
{\displaystyle \partial _{k}={\frac {\partial }{\partial x_{k}}}} define a holonomic basis of the tangent space. The collection of tangent spaces at all points...
on computer algebra, particularly on holonomicfunctions, with applications to combinatorics, special functions, knot theory, and physics. Together with...
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