Concise formulary
Some definitions and formulae from the main text are represented in concise form[1].
1 Q0-triangle
Let be j, m, n natural numbers, k an integer number with |k| greater or equal n and p contained in the interval [0,1]. We define
The function Q0P(n,k,p) represents the probability of reaching coordinate k after n steps of a Bernoulli random walk, if the probability of a step in positive k-direction (e.g. right hand direction) is equal to p (and so for a step to the opposite direction is equal to 1-p). The numbers Q0 (n, k) of the Q0-triangle correspond to the special case of same probabilities for steps to the right and to the left, i.e. for p=(1-p)=0.5:
The probabilities Q0Z (n) for return ("central meeting probabilities") correspond to the special case k = 0:
2 Q1-triangle
The Q1-triangle results from a superposition of two Q0-triangles with opposite sign, starting in position n=1, k=±1 after multiplication by 1/2. Addition of both means a "discrete differentiation" along k.
The absolute values |Q1 (n, k)| also arise, if after starting in row n=1 the following rows are constructed in usual way, but the numbers in the row centers k=0 are set to 0 in every row with even row number respectively, are so to speak "flown out", so that they can't be sources subsequently. Let for every even row number n be -Q2Z(n) "flowing out probability" there, i.e. the probability for flowing out centrally. Q2Z(n) is equivalent to the 1nd (discrete) derivative of Q1(n,k) in k=0 along k, i.e. Q2Z(n) = (Q1(n-1,1)-Q1(n-1,-1))/2; so Q2Z(n) is in k=0 the 2nd derivative of Q0(n,k) along k. It holds:
3 Q0M-triangle
Q0M(n,k) is in case of odd n antisymmetric and
in case of even n symmetric regarding to k=0 .
So addition behavior of right and left side is like the one of amplitudes of fermions resp. bosons.
4 Taylor series expansions
5 Limits
6 Multiple discrete differentiation (Formation of higher-order finite differences)
Similarly to the analytical case multiple discrete differentiation can be defined recursively (by formation of higher-order finite differences). Let be QDP(d,n,k) the d times along k differentiated function Q0P (n, k, p), then
and for n ³ d ³ 1
.
At this n ³ d is necessary that enough values are available to build a (finite) difference of d-th order.
6.1 Binomial coefficients and multiple differentiation (example matrix)
{BinCoeffDiffMatrix}The representation of operators as matrices is often useful in discrete considerations. Here a matrix representation of the operator for discrete differentiation in form of an example matrix with high number of dimensions for clarification of the development: Multiplication of a 15-dimensional vector by the following matrix
¦ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 ¦
¦ -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ¦
¦ 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 ¦
¦ 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 ¦
¦ 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 ¦
¦ 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 ¦
¦ 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 ¦
D := ¦ 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 ¦ * 1/2
¦ 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 ¦
¦ 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 ¦
¦ 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 ¦
¦ 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 ¦
¦ 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 ¦
¦ 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 ¦
¦ 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 ¦
means first order discrete differentiation "along" the index k of the vector components (calculation of the finite first order difference - the shift dk of the index k is 2, therefore division by 2). Multiplication by the n-ten power D^n of this matrix yields n-th order discrete differentiation (formation of the finite n-th order difference). For example means multiplication by
¦ -20 0 15 0 -6 0 1 0 0 1 0 -6 0 15 0 ¦
¦ 0 -20 0 15 0 -6 0 1 0 0 1 0 -6 0 15 ¦
¦ 15 0 -20 0 15 0 -6 0 1 0 0 1 0 -6 0 ¦
¦ 0 15 0 -20 0 15 0 -6 0 1 0 0 1 0 -6 ¦
¦ -6 0 15 0 -20 0 15 0 -6 0 1 0 0 1 0 ¦
¦ 0 -6 0 15 0 -20 0 15 0 -6 0 1 0 0 1 ¦
6 ¦ 1 0 -6 0 15 0 -20 0 15 0 -6 0 1 0 0 ¦
D = ¦ 0 1 0 -6 0 15 0 -20 0 15 0 -6 0 1 0 ¦ * 1/64
¦ 0 0 1 0 -6 0 15 0 -20 0 15 0 -6 0 1 ¦
¦ 1 0 0 1 0 -6 0 15 0 -20 0 15 0 -6 0 ¦
¦ 0 1 0 0 1 0 -6 0 15 0 -20 0 15 0 -6 ¦
¦ -6 0 1 0 0 1 0 -6 0 15 0 -20 0 15 0 ¦
¦ 0 -6 0 1 0 0 1 0 -6 0 15 0 -20 0 15 ¦
¦ 15 0 -6 0 1 0 0 1 0 -6 0 15 0 -20 0 ¦
¦ 0 15 0 -6 0 1 0 0 1 0 -6 0 15 0 -20 ¦
6-th order discrete Differentiation resp. calculation of the finite 6-th order difference. The rows resp. columns of the matrix D^n contain the binomial coefficients, divided by 2^n, in this example the numbers 6!/(k!·(6 - k)!·2^6) = Q0(6, 2k-6).
7 Special differences
horizontal (along localization):
vertical (along time):
Correspondence in the middle:
7.1 Schrödinger discretely
8 Scalar products
8.1 horizontal
8.2 vertical
.
.
8.3 Orthogonality after multiple discrete differentiation (analogously to Hermite polynomials)
Let be d,l ³ n.
We define the weighted scalar product QSP by
.
Then for d¹l is valid
(i.e. orthogonality) and otherwise
,
particularly
.
The denominator in the last expression corresponds to the number of the way possibilities from point (0,0) to point (n,n-2d) in the Q0-triangle.
9 Sums
10 Moments
10.1 vertical
10.2 horizontal
10.2.1 relative to the border
11 Sums and moments for variable p
This chapter contains some elementary formulae for variably p (and n>0).
The first finite difference (discrete derivative) of Q0P is
11.1 Sums
11.2 Deviation relative to the border
In the border the probabilities p and 1-p are very different. With p->0 also v->0 (low temperature).
11.3 Deviation relative to the origin
In the center the probabilities p and 1-p are nearly equal, i.e. p-> 1/2 and with that v-> C (the borderline case p=1/2 resp. v=C is represented by Q0 and Q1).
12 Analytic representations
Let be
i.e.
then is valid for every sequence (kn) with (kn)^3/n^2®0 für n®¥ (p. 80 [likr])
12.1 Schrödinger analytically
12.2 Behavior for n-> inf; Dirac delta-function
{DiracDeltaFu}It is
The behavior for n®¥ can be illustrated by a n proportional scale fitting, i.e. by a horizontal compression and a vertical stretching by respectively the factor n. This doesn't touch the value of the integral:
For n®¥ therefore the function f(x) = n Q0E(n, nx) / 2 converges towards the Dirac delta-function.
12.3 Multiple differentiation and Hermite polynomials
{HermPol}The Hermite polynomials Hn(x) are (except sign) special cases of pre-factors resulting from multiple differentiation:
