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COMMENTS
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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.
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 a(n)/n where the a(n)'s are A060872 or, equivalently, by b(n) of A038548.
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
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