Algebra and number theory: API index

Find an operation by its purpose. Entries link to the complete signature and contract.

HMath.Algebra

API	Purpose
Semiring	Commutative addition, associative multiplication, distributivity, additive identity and absorbing zero.
zero	Additive identity.
one	Additive identity.
plus	Additive identity.
times	Additive identity.
Ring	A semiring whose additive monoid is an abelian group.
negateRing	Additive inverse.
Field	A commutative ring in which each nonzero element has an inverse.
inverseField	Multiplicative inverse; zero returns a domain error.
Group	An associative operation with identity and inverse.
groupIdentity	Group identity.
groupCombine	Group identity.
groupInverse	Group identity.
Additive	A ring viewed as its additive group.
Additive	Declaration and type contract below.
getAdditive	Recover the underlying ring element.
Module	A left module with checked addition for runtime vector shapes.
moduleScale	Scalar action.
moduleAdd	Scalar action.
VectorSpace	A module over a field.
linearCombination	Checked linear combination.
semiringLaws	Exhaustive semiring identities on the supplied finite sample.
ringLaws	Semiring laws plus additive inverses on a finite sample.
fieldLaws	Ring laws, commutative multiplication and nonzero inverses.
groupLaws	Group identities on a finite sample.
moduleLaws	Check module scalar identity, composition, and both distributive laws.
Modular	A canonical residue in Z/mZ, retaining its runtime modulus.
modular	Construct a residue, requiring m>1; arbitrary signed values are reduced.
modularValue	Canonical value in [0,m).
modularModulus	Modulus greater than one.
addModular	Checked residue addition.
subtractModular	Checked residue subtraction.
multiplyModular	Checked residue multiplication.
inverseModular	Invert a unit; zero and nonunits are domain errors.
powerModular	Integer power, with negative powers supported for units.
FiniteField	An irreducible monic modulus over the proven prime field F_p.
finiteField	Construct GF(p^n) from an irreducible polynomial of positive degree.
primeField	Prime field GF(p), represented as F_p[x]/(x).
fieldCharacteristic	Prime characteristic p.
fieldDegree	Extension degree n.
fieldCardinality	Exact carrier size p^n.
fieldModulus	Ascending coefficients of the monic irreducible defining polynomial.
FieldElement	A canonical residue polynomial of degree smaller than its field degree.
fieldElement	Reduce an arbitrary integer polynomial into a finite field.
fieldElementCoefficients	Canonical ascending residue coefficients (trailing zeros removed).
fieldElements	Enumerate the complete carrier, rejecting sizes above an explicit storage budget.
addField	Addition; different defining fields are rejected even if isomorphic.
negateField	Additive inverse in the same field.
subtractField	Checked field subtraction.
multiplyField	Checked field multiplication, reduced modulo the defining polynomial.
inverseFieldElement	Nonzero inverse via the finite-field identity a^(q-2).
powerField	Binary exponentiation, with inverses for negative exponents.
finiteFieldLaws	Exhaustive carrier checks for small fields, including distributivity, associativity, identities and every nonzero inverse.

HMath.Algorithms

API	Purpose
lowerBound	Given a target and ascending list, return the first zero-based insertion index whose value is >= target, or length at the end.
upperBound	Given a target and ascending list, return the first insertion index whose value is > target.
binarySearch	Given target and ascending list, return Just the first matching zero-based index or Nothing if absent.
interpolationSearch	Search ascending Integer values using value-based interpolation, returning the first matching index or Nothing.
exponentialSearch	Given a target and ascending finite list, grow a search range then find a matching index or Nothing.
ternarySearch	Fixed-iteration minimization of a unimodal finite function on [lo,hi].
quickselect	Given a zero-based order-statistic index and list, return the element at that position in ascending order without returning a sorted list.
introselect	Like quickselect, but bound pivot depth and fall back to heap sorting.
partialSort	Given k in [0,length] and a list, return the k smallest values in ascending order, retaining duplicates.
mergeSort	Stable ascending merge sort under a lawful total Ord; accepts empty lists.
quicksort	Ascending reference quicksort preserving equal values, using the head as pivot.
heapsort	In-place-style array max heap with persistent updates.
introsort	Ascending comparison sort with bounded quickselect-style partition depth and heap-sort fallback.
countingSort	Given a nonnegative value-range budget then an Int list, return ascending counting sort.
radixSort	Sort unbounded Integers in ascending order by radix digits, separating negative and nonnegative magnitudes.
bucketSort	Given bucket count in [1,65536] and finite Doubles in [0,1], sort ascending using buckets and merge sort.
argsort	Return original zero-based indices in stable ascending value order.
topK	Alias of partialSort: return the k smallest values in ascending order, not the k largest. k outside [0,length] fails.
quantileSelect	Given finite q in [0,1] and a nonempty finite Double list, linearly interpolate the ascending order statistics at q*(n-1).
deduplicate	Sort ascending and keep one representative of each equal value.
numericGroups	Sort ascending and return each distinct value paired with its frequency.
runLengthEncode	Encode adjacent equal runs as (value,count), preserving run order without sorting.
runLengthDecode	Given maximum output length then (value,count) runs, expand runs.
prefixSums	Return cumulative sums including a leading zero; output length is n+1.
suffixSums	Return suffix sums including a final zero; output length is n+1.
differenceArray	Return first differences relative to a leading zero, preserving length.
applyDifferences	Reconstruct values from first differences by inclusive cumulative sum.
coordinateCompression	Return sorted unique keys and zero-based ranks for the original input order.
intervalSweep	Half-open interval union length and peak overlap; zero-length intervals contribute neither.
divideConquerDP	One DP layer next[j] = min_{0<=k<j} previous[k]+cost k j.
tabulateDP	Bottom-up bounded DAG recurrence; dependencies must have lower indices.
memoizedDP	Top-down memoization with a node budget.
binaryLift	K-fold application of a total finite successor map, by repeated squaring.
integerMatrixPower	Given square Integer rows and a nonnegative exponent, return exact matrix exponentiation by squaring; exponent zero returns identity.
polynomialHash	Given base, modulus >1, then Integer sequence, fold h=(h*base+x) mod modulus from zero.
rollingHashes	Noncryptographic polynomial rolling hash; equal hashes can collide.

HMath.Combinatorics

API	Purpose
permutationsCount	Given n,k in [0,4096], return n!/(n-k)! exactly, or zero when k>n.
combinationsCount	Given n,k in [0,4096], return the exact binomial coefficient, or zero when k>n.
multinomial	Given nonnegative group sizes whose sum is <=4096, return factorial(total)product(factorial sizes).
permutationsOf	Distinct positions, including when values compare equal.
combinationsOf	Given output budget, selection length k, then list, enumerate position-distinct k-element combinations in source order.
rankPermutation	A permutation of [0,n), in lexicographic Lehmer order.
unrankPermutation	Given size n in [0,4096] then zero-based rank in [0,n!), return the lexicographic permutation of [0..n-1].
rankCombination	Given universe size n then strictly increasing distinct indices in [0,n), return the zero-based lexicographic combination rank.
unrankCombination	Given universe size n, selection size k, then zero-based rank, return the increasing index list.
derangementCount	Return exact number of permutations of n elements having no fixed points.
derangements	Enumeration uses the factorial upper bound, so a request may be rejected even if its filtered derangement output would fit.
integerPartitions	Nonincreasing positive parts.
setPartitions	Set partitions of [0,n), using restricted-growth order of block labels.
stirlingFirst	Signed first-kind Stirling numbers, coefficients of the falling factorial.
stirlingSecond	Given n,k, return the exact number of partitions into k nonempty unlabeled blocks.
bellNumber	Return the exact number of set partitions of n labeled objects, summing second-kind Stirling numbers.
eulerianNumber	Given n,k, return the Eulerian recurrence counting permutations by ascents.
lahNumber	Given n,k in [0,4096], return unsigned Lah number n!/k!*choose(n-1,k-1), with (0,0)=1 and impossible positive-size partitions giving zero.
catalanNumber	Return exact Catalan number choose(2*n,n)/(n+1), requiring n in [0,4096].
fibonacci	Return exact zero-based Fibonacci number by fast doubling for n in [0,4096].
lucas	Return zero-based Lucas number using neighboring Fibonacci numbers. n and n+1 must both satisfy the [0,4096] bound, so n<=4095.
pascalTriangle	Return Pascal rows zero through n inclusive, starting [[1]]. n must lie in [0,512]; invalid orders fail.
inclusionExclusion	Union cardinality from all nonempty intersection sizes, indexed by subset bit mask 1 through 2^n-1.
subsets	Given maximum output count then list, enumerate its position-based powerset.
cartesianProduct	Given output budget then a list of lists, return all ordered choices using sequence.
multisetPermutations	Given budget then ordered values, enumerate distinct multiset permutations in ascending lexicographic order.
multisetCount	Return the multinomial count of distinct permutations of a list, grouping equal values in sorted order.
linearRecurrence	a[n] = sum c[i]*a[n-i-1]; initial values have the same length as coefficients.
generatingCoefficients	Formal power-series division P(x)/Q(x) around zero, exact rationals.
convolveSeries	Given output length n then two coefficient lists, return the first n exact Integer convolution coefficients, treating absent coefficients as zero. n must lie in [0,4096].
finiteDifferences	Return successive forward-difference rows beginning with the original list and ending at the last singleton.
finiteSum	Sum a finite list using its Num arithmetic.

HMath.Complex

API	Purpose
Complex	A data type representing complex numbers.
complex	Construct a finite number from real and imaginary parts.
realPart	Extracts the real part of a complex number.
imagPart	Extracts the imaginary part of a complex number.
magnitude	The non-negative magnitude of a complex number.
phase	The phase of a complex number, in the range (-pi, pi].
conjugate	The conjugate of a complex number.
reciprocal	Reciprocal using scaled division to avoid squaring a large norm.
add	Checked sum.
subtractComplex	Checked difference.
multiply	Checked product.
divide	Smith scaled division.
power	Principal complex power. 0^0 is 1; other zero-base exponents must be positive real.
Elementary	Elementary functions with principal inverse-function branches.
Exp	Declaration and type contract below.
Log	Declaration and type contract below.
Sqrt	Declaration and type contract below.
Sin	Declaration and type contract below.
Cos	Declaration and type contract below.
Tan	Declaration and type contract below.
Sinh	Declaration and type contract below.
Cosh	Declaration and type contract below.
Tanh	Declaration and type contract below.
Asin	Declaration and type contract below.
Acos	Declaration and type contract below.
Atan	Declaration and type contract below.
Asinh	Declaration and type contract below.
Acosh	Declaration and type contract below.
Atanh	Declaration and type contract below.
elementary	Apply an elementary function, rejecting poles and nonfinite results.
fromPolar	Polar construction with nonnegative finite radius.
toPolar	Radius and principal phase; the phase of zero is defined by Data.Complex.
rotate	Counterclockwise rotation in the complex plane.
complexMatrix	Checked finite complex matrix, at most 256 rows and columns.
multiplyMatrices	Matrix multiplication with finite-result validation.
conjugateTranspose	Hermitian transpose with finite-input validation.
complexFFT	Existing checked mixed-radix/Bluestein forward FFT.
complexIFFT	Existing checked inverse FFT, normalized by sample count.

HMath.NumberTheory

API	Purpose
gcdInteger	Nonnegative greatest common divisor; gcdInteger 0 0 = 0.
lcmInteger	Nonnegative least common multiple; zero if either argument is zero.
extendedGCD	(g,x,y) with ax+by=g>=0, also for negative inputs and zero.
bezout	Bezout coefficients only; see extendedGCD.
modPow	Binary modular exponentiation.
modInverse	Canonical inverse residue in [0,m).
chineseRemainder	Generalized CRT for (residue,positive modulus) pairs.
linearDiophantine	One solution and a direction: (x,y,dx,dy) represents every solution (x+kdx,y+kdy) of ax+by=c.
integerSquareRoot	Floor of the square root, computed by exact integer Newton iteration.
perfectPower	Find a representation n = base^exponent with exponent greater than one.
isPrimeExact	Proven primality using trial division.
nextPrime	First proven prime at least the requested integer.
millerRabin	Strong Miller--Rabin for explicitly supplied bases (each >=2).
bailliePSW	Baillie--PSW: base-2 strong Miller--Rabin plus strong Lucas--Selfridge.
sieveEratosthenes	Eratosthenes sieve through an inclusive nonnegative upper bound.
sieveAtkin	Atkin quadratic-residue sieve, followed by removal of prime-square multiples.
segmentedSieve	Segmented sieve on the inclusive interval [lo,hi].
primeFactorization	Exact trial factorization of a positive integer.
pollardRho	Pollard rho with Floyd cycle detection, explicit seed and iteration budget.
pollardPMinusOne	Pollard p-1 stage one with a smoothness bound >=2 and base.
fermatFactor	Fermat difference-of-squares factorization, with an explicit step budget.
divisors	All positive divisors in increasing order, under a factorization budget.
divisorSum	Sum of positive divisors.
divisorCount	Number of positive divisors.
eulerTotient	Euler totient, including phi(1)=1.
carmichael	Carmichael's exponent of the multiplicative unit group.
mobius	Mobius function on positive integers.
liouville	Liouville function, counting prime factors with multiplicity.
multiplicativeOrder	Multiplicative order of a unit modulo a modulus >1.
primitiveRoot	Smallest generator of the units modulo m, if that group is cyclic.
discreteLog	Baby-step giant-step discrete logarithm of a unit.
legendreSymbol	Legendre symbol for a proven odd prime modulus, with proof budget.
jacobiSymbol	Jacobi symbol for a positive odd denominator.
kroneckerSymbol	Kronecker symbol, extending Jacobi to every integer denominator.
tonelliShanks	Tonelli--Shanks modular square root for a proven prime modulus.
continuedFraction	Finite simple continued fraction of a rational, using floor division.
fromContinuedFraction	Evaluate a finite continued fraction.
convergents	Successive convergents, rejecting invalid finite prefixes.
farey	Farey sequence of positive order, including zero and one.
pellFundamental	Minimal positive solution to x^2-d*y^2=1 via continued fractions of sqrt d.
isPerfectNumber	Perfect-number predicate on positive integers.
areAmicable	Amicable-number predicate; equal numbers are not an amicable pair.
partitions	Integer partitions in descending part order.
partitionCount	Partition count by exact dynamic programming, O(n^2) work and O(n) storage.
fibonacciFast	Signed-index Fibonacci by fast doubling; F(-n)=(-1)^(n+1)F(n).

HMath.Polynomial

API	Purpose
Polynomial	Canonical, finite ascending coefficient sequence.
polynomial	Construct a polynomial, deleting trailing exact zeros (no floating tolerance).
coefficients	Ascending coefficients, with no trailing zero; zero gives an empty list.
polynomialDegree	Degree of a nonzero polynomial; zero has no degree.
evaluatePolynomial	Horner evaluation, with the constant coefficient first in storage.
addPolynomial	Coefficientwise addition with canonical exact-zero normalization.
subtractPolynomial	Coefficientwise subtraction.
scalePolynomial	Multiply every coefficient by a scalar.
multiplyPolynomial	Exact schoolbook convolution, including zero polynomials.
composePolynomial	Composition p(q(x)) by polynomial Horner evaluation.
dividePolynomial	Long division over a field.
monicPolynomial	Normalize a nonzero polynomial to leading coefficient one; zero remains zero.
gcdPolynomial	Monic Euclidean GCD over an exact field; gcd(0,0)=0.
differentiatePolynomial	Formal derivative.
integratePolynomial	Formal integral with the supplied integration constant.
squareFreeFactors	Monic square-free factors with positive multiplicities over Rational.
rationalRoots	Rational roots with exact multiplicities using the rational-root theorem.
factorRationalPolynomial	Complete rational factorization by bounded Kronecker interpolation.
sturmSequence	Sturm chain over exact rationals, after removing the derivative GCD.
countRealRoots	Number of distinct real roots in (a,b].
isolateRealRoots	Isolate every distinct real root in rational intervals with disjoint interiors and width at most tolerance.
durandKerner	Durand--Kerner simultaneous complex root iteration.
aberthEhrlich	Aberth--Ehrlich simultaneous iteration, combining Newton corrections with mutual-root repulsion.
companionMatrix	Frobenius companion matrix of a nonconstant polynomial, with ones on the subdiagonal and negated monic coefficients in the last column.
lagrangeInterpolation	Exact Lagrange interpolating polynomial.
newtonInterpolation	Newton divided differences, returned in the standard power basis.
chebyshevPolynomial	Chebyshev T_n, normalized by T_0=1, T_1=x.
legendrePolynomial	Legendre P_n, normalized by P_n(1)=1.
hermitePolynomial	Physicists' Hermite H_n: H_0=1, H_1=2x.
laguerrePolynomial	Ordinary Laguerre L_n with alpha=0 and L_n(0)=1.
bernsteinPolynomial	Bernstein basis binomial(n,k)x^k(1-x)^(n-k) for 0<=k<=n.
polynomialRegression	Least-squares power-basis regression using column-pivoted Householder QR.
chebyshevApproximation	Truncated Chebyshev interpolation at first-kind nodes on [-1,1], returned in power basis.
PowerSeries	A finite Taylor jet, coefficients ascending and zeros retained.
powerSeries	Construct a jet with explicit coefficients, including trailing zeros.
seriesCoefficients	Stored Taylor coefficients; coefficient n multiplies x^n, not x^n/n!.
addSeries	Add jets to their common truncation order.
multiplySeries	Multiply jets by truncated convolution.
reciprocalSeries	Reciprocal jet from coefficient recurrence; constant term must be nonzero.
divideSeries	Divide jets to their common order; denominator constant must be nonzero.
composeSeries	Formal composition to the inner jet's order.
differentiateSeries	Derivative jet has one fewer coefficient.
integrateSeries	Integral jet with supplied constant, one coefficient longer.
expSeries	Exponential of a finite jet from b'=a'b; constant uses the coefficient type's exp.
logSeries	Formal log through derivativedivisionintegration.
RationalFunction	Reduced exact rational function with monic denominator.
rationalFunction	Normalize by polynomial GCD and reject a zero denominator.
rationalNumerator	Reduced numerator.
rationalDenominator	Monic reduced denominator.
evaluateRationalFunction	Exact evaluation, rejecting poles.
addRationalFunction	Exact rational-function addition with cancellation.
multiplyRationalFunction	Exact rational-function multiplication with cancellation.
partialFractions	Exact partial fractions over Q.

HMath.Special

API	Purpose
SFResult	Finite value and estimated absolute numerical error.
SFResult	Declaration and type contract below.
sfValue	Returned special-function numerical estimate.
sfError	Absolute numerical error estimate accompanying sfValue.
gamma	Real gamma using C99 libm; rejects nonfinite arguments and nonpositive integer poles.
logGamma	Natural logarithm of gamma for strictly positive finite arguments; avoids overflow of gamma itself.
gammaP	Regularized lower incomplete gamma.
gammaQ	Regularized upper gamma Q(a,x); the upper-tail continued fraction is evaluated directly when x>=a+1.
incompleteGamma	Unregularized lower gamma gamma(a,x)=Gamma(a)*P(a,x).
beta	Beta function for strictly positive parameters, evaluated through log gamma.
logBeta	Log beta for strictly positive finite parameters.
betaI	Regularized lower incomplete beta I_x(a,b), positive parameters and x in [0,1], via reflected Lentz continued fraction.
incompleteBeta	Unregularized lower incomplete beta B_x(a,b).
digamma	Log-gamma derivative for x>0, using recurrence to x>=12 and an inverse-power asymptotic expansion.
trigamma	First polygamma derivative for x>0, using the Hurwitz-zeta identity.
polygamma	Derivative order 0..32 of digamma for x>0; higher orders use signed factorial times Hurwitz zeta.
erf	Real error function from C99 libm; finite arguments required.
erfc	Real complementary error function from C99 libm; upper tails avoid subtracting erf from one.
inverseErf	Real inverse error function for -1<x<1, using the normal inverse identity.
erfcx	exp(x^2)*erfc(x), with a large-positive-x inverse-power asymptotic expansion to avoid overflow.
normalCDF	Standard-normal CDF through erfc, with stable small lower tails.
normalQuantile	Standard-normal inverse CDF for 0<p<1.
dawson	Dawson integral for |x|<=100; scaled quadrature below |x|=10, inverse-power asymptotics above.
exponentialIntegral	E_n(x), integer n=0..100000,x>0.
logarithmicIntegral	Real principal-value Ei for x>0; logarithmic integral li(x)=Ei(log x), with the real PV continuation on 0<x<1.
sineIntegral	Real Si(x) on |x|<=1000, with its removable singularity explicitly filled at zero.
cosineIntegral	Real Ci(x) on 0<x<=1000, using phase-resolved quadrature near zero and inverse-power oscillatory asymptotics above x=50.
fresnel	Fresnel (C,S), phase pi*t^2/2, |x|<=100.
ellipticF	Legendre incomplete elliptic integrals, parameter m (not modulus k), real branch |phi|<=pi/2, m<1; Pi additionally n<1.
ellipticE	Legendre incomplete second-kind elliptic integral with the same amplitude/parameter convention.
ellipticPi	Legendre third-kind integral with arguments characteristic n, amplitude phi, parameter m; n<1, m<1, |phi|<=pi/2.
jacobiElliptic	Real (sn,cn,dn), |u|<=10^6 and 0<=m<1, using periodic amplitude inversion.
besselJ	Integer-order J_n(x) from libm, |n|<=100000 and finite real x.
besselY	Integer-order Y_n(x) from libm, |n|<=100000 and x>0.
besselI	Modified Bessel I_n(x) via its convergent series, integer order 0..100 and |x|<=700.
besselK	Modified Bessel K_n(x)=exp(-x)*scaledBesselK(n,x), integer order 0..100,x>0.
sphericalBesselJ	Spherical j_n(x), order 0..100; uses a series for small x relative to order and upward recurrence otherwise.
sphericalBesselY	Spherical y_n(x), integer order 0..100,x>0, by upward recurrence.
hankel	Complex H_n^(kind)(x)=J_n(x)+/-iY_n(x), kind 1 or 2, integer order and x>0.
airy	Airy Ai,Bi and derivatives solve y''=x y by convergent Maclaurin series.
struve	Real Struve H_nu(x) by a convergent series; nu>-3/2 and 0<=x<=30.
hypergeometric	Generalized pFq power series.
hypergeometric1F1	Kummer confluent hypergeometric 1F1(a;b;z), real power-series evaluation with denominator-pole rejection.
hypergeometric2F1	Gauss 2F1(a,b;c;z) in the real power-series disk |z|<1.
complex1F1	Complex Kummer 1F1 in its entire z plane, rejecting denominator poles and nonfinite arithmetic; cancellation can limit accuracy for large imaginary z.
legendre	Legendre P_n(x), n=0..1000 and |x|<=1.
associatedLegendre	Associated P_l^m(x) with Condon--Shortley phase, 0<=m<=l<=1000 and |x|<=1.
sphericalHarmonic	Orthonormal complex spherical harmonic Y_l^m(theta,phi), theta in [0,pi], including negative-m conjugation symmetry.
riemannZeta	Real zeta: Euler--Maclaurin for s>1, Euler-transformed eta for 0<s<1 and reflection for -170<s<0.
hurwitzZeta	Hurwitz zeta for s>1,q>0 using 64 explicit terms plus seven Euler--Maclaurin corrections.
polylogarithm	Real polylog Li_s(z) in the convergent power-series disk |z|<1; no cross-cut continuation is implied.
dilogarithm	Real Li_2(x) for x<=1, with reflection and negative-axis transformations for convergent evaluation.
lambertW	Real Lambert W branch 0 or -1, x>=-1/e and x<0 on branch -1, using bounded Halley iteration.
debye	Normalized Debye D_n(x), integer n=1..16,x>=0, with D_n(0)=1.
clausen	Periodic Clausen Cl_2(theta) from an integral with a squared-variable substitution at its logarithmic endpoint.
fermiDirac	Complete normalized Fermi--Dirac integral F_order(eta), 0<=order<=20, |eta|<=100.
boseEinstein	Complete normalized Bose--Einstein integral, same order range, -100<=eta<=0 and positive order at eta=0.
coulombWave	Regular and irregular Coulomb waves F_l,G_l.
gegenbauer	Gegenbauer C_n^alpha(x), degree 0..10000 and alpha>-12, alpha=0, by three-term recurrence.
parabolicCylinder	Real D_nu(x) via Kummer's representation; reference |x|<=6, |nu|<=10.
mathieu	Mathieu ce_n (even=True) or se_n, from a 32-term Fourier Galerkin eigenproblem. ce(0)=1 or se'(0)=1.
bernoulliPolynomial	Exact rational Bernoulli/Euler coefficients evaluated as Double.
eulerPolynomial	Euler E_n(x), n=0..100, from the Bernoulli duplication identity.
wigner3j	Angular momenta and projections are encoded as doubled integers, allowing half-integers exactly.
wigner6j	Racah factorial sum for a Wigner 6j symbol; each angular momentum is a doubled integer in 0..60.
wigner9j	Wigner 9j as a finite sum of three 6j symbols; accepts nine row-major doubled angular momenta in 0..30.
clebschGordan	Clebsch--Gordan coefficient <j1 m1,j2 m2|j m> through the Condon--Shortley 3j convention; all arguments doubled integers.
scaledBesselI	exp(-|x|)*I_n(x).
scaledBesselK	exp(x)*K_n(x) from its cosh integral, integer order 0..100,x>0; scaling is inside the integrand.