OFFSET
0,6
COMMENTS
Coefficients may be generated from a modified Riordan array (MRA) formed from Rgf(z,t) = (t/(1+z))/(exp(-z/(1+z))-t) with each row of the array acting to generate the succeeding polynomial P(n,t) from the preceding n polynomials.
The MRA is constructed by appending an n! to the left of the n-th row of the Riordan array A129652 and removing the unit diagonal.
The MRA is partially
1;
1, 1;
2, 3, 2;
6, 13, 9, 3;
24, 73, 52, 18, 4;
120, 501, 365, 130, 30, 5;
720, 4051, 3006, 1095, 260, 45, 6;
For the MRA:
1) First column is the n!'s.
2) Second column is A000262.
Then, e.g., from the terms in the MRA,
P(0,t) = 0!*(t-1)^0 = 1 from the n=0 row,
P(1,t) = 1!*(t-1)^1 + 1*P(0,t) = t from the n=1 row,
P(2,t) = 2!*(t-1)^2 + 3*P(0,t)*(t-1)^1 + 2*P(1,t)
P(3,t) = 3!*(t-1)^3 + 13*P(0,t)*(t-1)^2 + 9*P(1,t)*(t-1)^1 + 3*P(2,t)
generating
P(0,t) = (1)
P(1,t) = (0, 1)
P(2,t) = (-1, 1, 2)
P(3,t) = (4, -14, 10, 6) = 4 + -14 t + 10 t^2 + 6 t^3
P(4,t) = (-15, 83, -157, 89, 24)
P(5,t) = (56, -424, 1266, -1724, 826, 120)
P(6,t) = (-185, 1887, -8038, 17642, -19593, 8287, 720)
P(7,t) = (204, -4976, 36226, -126944, 239576, -234688, 90602, 5040)
For the polynomial array:
1) The first column is A009940 = (-1)^n * n!*Lag(n,1) =(-1)^n* n!* Lag(n,-1,-1).
2) Row sums are n!.
3) Highest order coefficient is n!.
4) Alternating row sum is below.
Then, with Rf(n,t) = [ t/(1-t)^(n+1) ] * P(n,t)/n!, the polylogs are given umbrally by
Li(-n,t)/n! = [ 1 + Rf(.,t) ]^n for n = 0,1,2,... so conversely
Rf(n,t) = {[ Li(-(.),t))/(.)! ]-1}^n.
Note umbrally [ Rf(.,t) ]^n = Rf(n,t) and
(1+Rf)^0 = 1^0 * [ Rf(.,t) ]^0 = Rf(0,t) = t/(1-t) = Li(0,t).
More generally, Newton interpolation holds and for Re(s) >= 0,
Li(-s,t)/(s)! = [ 1 + Rf(.,t) ]^s, when convergent in t.
Alternatively, the Rf's may be formed through differentiation of their o.g.f., the Rgf(z,t) above, which may also be written as
Rgf(z,t) = Sum_{k>=1} [ t^k ] * exp[ k * z/(z+1) ]/(z+1)
= Sum_{n>=0} [ (-z)^n ] * Sum_{k>=1} [ (t^k * Lag(n,k) ]
= Sum_{k>=0} [ (-z)^k ] * Lag(k,Li(-(.),t))
= Sum_{k>=0} [ z^k ] * {[ Li(-(.),t))/(.)! ]-1}^k
= exp[ Li(-(.),t)*z/(1+z) ]/(1+z),
and operationally as
Rgf(z,t) = {Sum_{k>=0} (-z)^k * Lag(k,tD)} [ t/(1-t) ]
= {Sum_{k>=0} (-z)^k * Lag(k,T(.,:tD:))} [ t/(1-t) ]
= {Sum_{k>=0} (-z)^k * Sum_{j>=0} Lag(k,j) (tD)^j /j!} [ x/(1-x) ]
where D is w.r.t. x at 0
= {Sum_{k>=0} (-z)^k*Sum_{j=0..k} (-1)^j*[ 1-Lag(k,.) ]^j*(:tD:)^j/ j!} [ t/(1-t) ]
= {Sum_{k>=0} (-z)^k * exp[ -[ 1-Lag(k,.) ]* :tD: ]} [ t/(1-t) ]
where (:tD:)^n = t^n * D^n, D is the derivative w.r.t. t unless otherwise stated, Lag(n,x) is a Laguerre polynomial and T(n,t) is a Touchard / Bell / exponential polynomial.
Hence [ t/(1-t)^(n+1) ] * P(n,t)/n! = Rf(n,t)
= {Sum_{k=0..n} (-1)^n-k)*[ C(m,k)/k! ]*(tD)^k} [ t/(1-t) ]
= {Sum_{k=0..n} (-1)^(n-k)*[ C(m,k)/k! ]*Sum_{j=0..k} S2(k,j)*(:tD:)^j} [ t/(1-t) ]
= {Sum_{k>=0} (-1)^(n-k) * Lag(n,k) * (tD)^k/k!} [ x/(1-x) ] where D is w.r.t. x at 0
= {Sum_{k=0..n} (-1)^(n-k)* [ 1-Lag(n,.) ]^k *(:tD:)^k/k!}[ t/(1-t) ],
where S2(k,j) are the Stirling numbers of the second kind and C(m,k), binomial coefficients.
The P(n,t) are related to the Laguerre polynomials through
P(n,t) = (-1)^n n! [ (1-t)^(n+1)} ] Sum_{k>=0} [ (t^k*Lag(n,k+1) ] = Sum_{m=0..n} a(n,m) * t^m
where a(n,m) = (-1)^n n! [ Sum_{k=0..m} (-1)^k * C(n+1,k) *Lag(n,m-k+1) ] .
Conjecture for the polynomial array:
The greatest common divisor of the coefficients of each polynomial is given by A060872(n)/n or, equivalently, by A038548(n).
Some e.g.f.'s for the Rf's are
exp[ -Rf(.,t)*z ] = exp{[ 1-Li(-(.),t)/(.)! ]*z}
= Sum_{n>=0} { (z^n/n!) * Sum_{k>=1} [ t^k * Lag(n,k) ] }
= Sum_{k>=1} { t^k * (e^z) * J_0[ 2*sqrt(k*z)}
= Sum_{n>=0} {(-1)^n*(z^n/n!)*(z^/j!)*Lag(n,-1)*Sum_{k>=1} [ t^k*k^n*(k+1)^j ]}
where J_0(x) is the zeroth Bessel function of the first kind.
The expressions (:tD:)^j}[ t/(1-t) ] and the Laguerre polynomials are intimately connected to Lah numbers and rook polynomials.
Some interesting relations to physics, probability and number theory are, for abs(t) < 1 and abs(z) < 1 at least,
BE(t,z) = Sum_{k>=0} [ (-z)^k ] *[ 1 + Rf(.,t) ]^k
= Rgf(-z/(1+z),t)/(1+z) = t/{exp(z)-t}, a Bose-Einstein distribution,
FD(t,z) = Sum_{k>=0} [ (-z)^k+1 ] *[ 1 + Rf(.,-t) ]^k
= -Rgf(-z/(1+z),-t)/(1+z) = t/{exp(z)+t}, a Fermi-Dirac distribution
and as t tends to 1 from below, z*BE(t,z) tends to the Bernoulli e.g.f., which is related by the Mellin transform to (s-1)!*Zeta(s). Taking Mellin transforms of BE and FD w.r.t. z gives the polylogarithm over different domains.
Since BE(2,z) is essentially the e.g.f. for the ordered Bell numbers, it follows that umbrally
n! * Lag(n,OB(.)) = P(n,2) and
n! * Lag(n,P(.,2)) = OB(n)
where OB(n) are the signed ordered Bell/Fubini numbers A000670.
I.e., P(n,2) and the ordered Bell numbers form a reciprocal Laguerre combinatorial transform pair,
or, equivalently, P(n,2)/n! and OB(n)/n! form a reciprocal finite difference pair, so
P(n,2)/n! = (-1)^(n+1) * Rf(n,2) = -{1-[ Li(-(.),2))/(.)! ]}^n and
OB(n) = -Li(-n,2).
Note that n!*Lag(n,(.)!*Lag(.,x)) = x^n is a true identity for general Laguerre polynomials Lag(n,x,a) with a = -1,0,1,..., so one could look at analogous higher-order reciprocal pairs containing OB(n).
In addition, a mixed-order iterated Laguerre transform gives
n!*Lag{n,(.)!*Lag[ .,P(.,2),0 ],-1}
= P(n,2) - n*P(n-1,2)
= n!*Lag[ n,OB(.),-1 ] = A084358(n), lists of sets of lists.
E(n,t)/n! = [ 1-t+P(.,t)/(.)! ]^n
P(n,t)/n! = [ E(.,t)/(.)!-(1-t) ]^n, or equivalently
[ E(.,t)/(1-t) ]^n = n!*Lag[ n,-P(.,t)/(1-t) ]
[ -P(.,t)/(1-t) ]^n = n!*Lag[ n,E(.,t)/(1-t) ], a Laguerre transform pair.
Then from known relations for the Eulerian polynomials, the alternating row sum of the polynomial array is
P(n,-1) = (-2)^(n+1) * n! * Lag[ n,c(.)*Zeta(-(.)) ]
where c(n) = [ 2^(n+1) - 1 ] and Zeta is the Riemann zeta function. And so
Zeta(-n) = n! * Lag[ n,-P(.,-1)/2 ] / [ 2 - 2^(n+2) ],
which also holds, with the summation limit of Lag extended to infinity, for n = s, any complex number with Re(s) > 0.
Then from standard formulas for the signed Euler numbers EN(n), the Bernoulli numbers Ber(n), the Genocchi numbers GN(n), the Euler polynomials EP(n,t), the Eulerian polynomials E(n,t), the Touchard / Bell polynomials T(n,t) and the binomial C(x,y) = x!/[ (x-y)!*y! ]
2^(n+1) * (1-2^(n+1)) * (-1)^n * Zeta(-n)
= 2^(n+1) * (1-2^(n+1)) * Ber(n+1)/(n+1)
= [ -(1+EN(.)) ]^n
= 2^n * GN(n+1)/(n+1)
= 2^n * EP(n,0)
= (-1)^n * E(n,-1)
= (-2)^n * n! * Lag[ n,-P(.,-1)/2 ]
= (-2)^n * n! * C{T[ .,P(.,-1)/2 ] + n, n}
= an integer = Q(n)
These are related to the zag numbers A000182 by Zag(n) = abs[ Q(2*n-1) ]. And, abs[ Q(2*n-1) ] / 2^q(n) = Zag(n) / 2^q(n) = A002425(n) with q(n) = A101921.
These may be generalized by letting n = s, a complex number, (or interpolating) to obtain generalized Laguerre functions or confluent hypergeometric functions of the first kind, M(a,b,x), or second kind, U(a,b,x), whose arguments are P(.,-1)/2, such as
E(s,-1)/[ 2^s*s! ] = -2*Li(-s,-1)/s! = (2-2^(s+2)) * Zeta(-s)/s!
= C{T[ .,P(.,-1)/2 ] + s, s} = Lag[ s,-P(.,-1)/2 ] = M[ -s,1,-P(.,-1)/2 ] or,
GN(s+1)/(s+1)! = EP(s,0)/s! = C{-T[ .,P(.,-1)/2 ]-1, n} = U[ -s,1,-P(.,-1)/2 ]/(s)!
And even more generally
E(s,t)/(1-t)^s = [ (1-t)/t ] Li(-s,t) = s!*Lag[ s,-P(.,t)/(1-t) ]
= s! * C{T[ .,P(.,t)/(1-t) ] + s, s} = s! * M[ -s,1,-P(.,t)/(1-t) ] .
The Laguerre polynomial expressions are fundamental as they can be interpolated to form general M[ a,b,-P(.,t)/(1-t) ] or U[ a,b,-P(.,t)/(1-t) ] which can then be related either directly or by binomial transforms to many important Sheffer sequences, not to mention group theory and Riemann surfaces.
Note for frequently occurring expressions above: The Laguerre polynomials of order -1 and 0 are intimately connected to Lah numbers and rook polynomials and (tD)^n [t/(1-t)] = T(n,:tD:) [t/(1-t)] generates an Eulerian polynomial in the numerator of a rational function. - Tom Copeland, Sep 09 2008
The deformed Todd operator on p. 12 of Gunnells and Villegas is Td(a,D) = -D / (a*exp(-D) - 1) = [-D/(1-D)] * Rgf(D/(1-D), 1/a) = -D * BE(1/a,-D) = D * FD(-1/a,-D), where BE and FD are the Bose-Einstein and Fermi-Dirac distributions given above. See also connections among the Eulerian polynomials, Ehrhart polynomials, and the Todd operator in Beck and Robins, especially pages 31 and 37. - Tom Copeland, Jun 20 2017
REFERENCES
M. Beck and S. Robins, Computing the Continuous Discretely, illustrated by D. Austin, Springer, 2007.
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
P. Gunnells and F. Villegas Lattice polytopes, Hecke operators, and the Ehrhart polynomial, arXiv:math/0405573 [math.CO], 2004.
FORMULA
a(n,m) = (-1)^n*n!*Sum_{k=0..m} (-1)^k*C(n+1,k)*Lag(n, m-k+1).
MATHEMATICA
a[n_, m_] := (-1)^n *n!*Sum[(-1)^k*Binomial[n+1, k]*LaguerreL[n, m-k+1], {k, 0, m}]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 23 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Tom Copeland, Sep 17 2007, Sep 27 2007, Sep 30 2007, Oct 01 2007, Oct 08 2007
EXTENSIONS
A173018 given as reference for Eulerian polynomials and typo in a Laguerre function corrected by Tom Copeland, Oct 02 2014
STATUS
approved