We can transform functions in the summands of an Engel expansion to sequence terms and or generating functions. A list gives E\left[{G(1+n)G(2+n)(2,2)_n}\right] \to A273935 = n!(n-1)!(2^n-1) \\ E\left[{G(1+n)G(2+n)}\right] \to A010790 = n!(n-1)! \\ E\left[{G(1+n)G(3+n)}\right] \to A129464 = -n(n+1)(n-1)!^2 \\ E\left[ 2^{-n^2-n+{24}} \pi ^{{2}+{4}} G(n+3)}{A^{3/2} G\left(n+{2}\right)}\right] = C(n) = {(n+1)!n!} \\ E\left[{\Gamma(n+1)}\right] \to 1,1,2,3,4,5,6,7,8,9, \cdots \,n \\ E\left[{\Gamma(n+1)^2}\right] \to 1,1,4,9,16,25,36,49,\cdots \,n^2 \\ E\left[{G(1+n)}\right] \to 1,1,1,2,6,24,120,720, \cdots \, n! \\ this is generated by the product of the reciprocals of the sequence terms ^n {a(k)} = E^{-1}(a(n))

Consider a transform that acts on a function to extract the product series coefficient. For example if a function allows a series expansion f(x) = ^\infty (1 + a_k x^k) = ^\infty b_k x^k then we define the transform K of f to be K_x[f](s) = a_s likewise the inverse transform gives K_s^{-1}[a_s](x) = f(x) then by definition of the Pochhammer-q function we have K_x[{2}(-1,x)_{\infty}] = 1 \\ K_x[(q,q)_{\infty}] = -1 then certain functions with number theoretic potential can be defined by the inverse transform K_s^{-1}[\lambda(s)] \\ K_s^{-1}[\mu(s)] \\ K_s^{-1}[\phi(s)] \\ K_s^{-1}[\Omega(s)] \\ for functions λ(s),μ(s),.. the Liouville function, Moebius function, totient function etc. If we attempt to extract the coefficeints of a well known function, for example \exp(x) = (1+1x)(1+{2})(1-{3})(1+{8})(1-{5})(1+{72})\cdots we find that the coefficients of odd prime powers p are −1/p. We find that _x\left[{1-x}\right](s) = (s) which is the characteristic function of non-zero powers of 2, which comes from the interpretation of that number of ways to partition each number into powers of 2 is 1. We have b_7 = (a_1a_2a_4 + a_3a_4 + a_2a_5 + a_1a_6 + a_7) which is clearly a sum over the ways to make 7 from unique choices of k. Whereas terms like a_5 = b_5 - b_2 b_3 + b_1 b_2 b_2 - b_1 b_4 + b_1 b_1 b_3 - b_1 b_1 b_1 b_2 where the signs seem to correspond to the number of terms and repeats are now allowed. This gives b_n = \left(^n [k_1=n]a_{k_1}\right) + \left(^n^n [k_1+k_2=n]a_{k_1}a_{k_2}\right) + \cdots and then a_n = \left(^n [k_1=n]b_{k_1}\right) - \left(^n^n [k_1+k_2=n]g_{k_1 k_2}b_{k_1}b_{k_2}\right) + \cdots where the gkl correct for multiplicities.

Create a transform somewhat in analogy to the Mellin transform which to some extent extracts sequence coefficients. _s[f(x)](s) \approx \Gamma(s)\phi(-s) where f(x) = ^\infty {s!}\phi(s) x^s instead consider a transform ℐ[f(x)](s) such that [f(x)](s) \approx \Gamma(s)\chi(-s) where (x)}{x} = ^\infty {s!}\chi(s) x^s and example, the function f(x) = x + x^2 has inverse as series f^{-1}(x) = x - x^2 + 2 x^3 - 5 x^4 + \cdots (x)}{x} = 1 - x + 2 x^2 - 5 x^3 + \cdots = ^\infty {s!} C_s x^s = ^\infty {s!} {(s+1)!} x^s then \chi(s) = {\Gamma(2+s)} then [x+x^2](s) ={\Gamma(2-s)} x^{-1}[[x+x^2](s)](x) = f^{-1}(x) RESULTS Then it would seem that [2x^2 - ](s) ={\Gamma(1-s)} [W(x)](s) =\Gamma(s)(-1)^s \\ [-W(-x)](s) =\Gamma(s) \\ \left[{1-x}\right](s) =\Gamma(s)\Gamma(1-s) \\ \left[-2x}{2x}\right](s) =\Gamma(s)\Gamma(2-s) \\ \left[\log\left({x}\right)\right](s) =\Gamma(s-1)\\ \left[W\left({x}\right)\right](s) =\Gamma(s-2)\\ \left[e^x-1\right](s) ={\Gamma(2-s)} \\ \left[\log(x)\right](s) =(-1)^{1-s} \Gamma(s-1) \\ \left[{e^x-1}\right](s) ={1-s} \\ \left[-x-W(-xe^{-x})\right](s) =\Gamma(s)\zeta(s) \\ Some more generalised ones [\log(x^k)](s) = \left(-{k}\right)^{1-s} \Gamma(s-1)\\ \left[W\left({x^k}\right)\right](s) =k^{-1-1/k+s}\Gamma(s-1-{k})\\ \left[W\left(x^k\right)\right](s) =(-k)^{-1+1/k+s}\Gamma(s-1+{k})\\ \left[-{x+W(-e^{-x}x)}\right](s) = {s}\\ \left[-2W(-}{2})\right](s) = s^2 \Gamma(s)\\ \left[{\log(k/x)}\right](s) = k \Gamma(1-s)\\ \left[{1 - W(ex/k)}\right](s) = k s \Gamma(1-s)\\ \left[-x^k W(}{A})\right](s) = -A x^{k s}\Gamma(s)\\ As described in a previous article on here: It would appear that for the function f(x)=x^m+x, m>1 we get a series g(x)=^\infty {n}}{(m-1)n+1} these then have a set of consistent, hypergeometric series explainable as g(x)=_{(m-1)}F_{(m-2)}\left(\left\{{m},{m},\cdots,{m}\right\};\left\{{m-1},\cdots,{m-1},{m-1}\right\};-}{(m-1)^{m-1}}\right)\cdot x then {x}=_{(m-1)}F_{(m-2)}\left(\left\{{m},{m},\cdots,{m}\right\};\left\{{m-1},\cdots,{m-1},{m-1}\right\};-}{(m-1)^{m-1}}\right) which would give [x+x^m](s) = _x[\;_{(m-1)}F_{(m-2)}\left(\left\{{m},{m},\cdots,{m}\right\};\left\{{m-1},\cdots,{m-1},{m-1}\right\};-}{(m-1)^{m-1}}\right) ](s) which gives [x+x^2](s) = {\Gamma(2-s)}\\ [x+x^3](s) = {2})\Gamma({2})}{2 \Gamma(2-s)} \\ [x+x^4](s) = {2}}\pi \Gamma(-4s/3)\Gamma(s/3)}{\Gamma(2/3-s/3)\Gamma(4/3-s/3)\Gamma(-s/3)} =?={3 \Gamma(2-s)} it then seems like {m-1}\right)\Gamma\left({m-1}\right)}{(m-1)\Gamma(2-s)} = _x[x+x^m](s) and more generally {m-1}}\Gamma\left(1-{m-1}\right)\Gamma\left({m-1}\right)}{(m-1)\Gamma(2-s)} = _x[x+a x^m](s) FURTHER SMALL POLYNOMIALS There are some other small polynomials that give integer sequences upon reversion. Consider x − x² − x³. ^{-1}_s[\Gamma(a+b s)](x) = (-1)^b(b+a)^b W(-{b+a}}x^{{b+a}}}{b+a})^b

ABSTRACT I introduce k-crossing paths and partitions and count the number of paths for each number of desired crossings k for systems with 11 points or less. I give some conjectures into the number of possible paths for certain numbers of crossings as a function of the number of points. INTRODUCTION A order n meandric partition is a set of the integers 1⋯n, such that a path from the south-west can weave through n points labeled 1⋯n without intersecting itself and finally heads east (examples are shown in Fig. 1). Counting the number of possible paths for n points is a tricky problem, and no recursion relation, generating function or explicit formula for the number of order n meandric partitions appears to have been found. This work is concerned with the number of paths that must intersect themselves exactly k times, where when k is 0, we have the meandric paths. It is possible to draw a line that deliberately crosses itself as many times as required, because of this we only consider a path to be k-crossing if k is the smallest number of crossings possible, that is a path that must cross itself k times (an example of a 3-crossing path over 9 points is given in Fig. 2). RESULTS Define ak(n) to be the number of configurations of n points where the path through them is forced to cross itself k times. For 0-crossings on n points we have the open meandric numbers, given in the OEIS as A005316 a_0(n) = 1, 1, 1, 2, 3, 8, 14, 42, 81, 262, 538, 1828, 3926, \cdots, \;\; n=0,1,\cdots this work has counted this for k > 0 by calculating all n! permutations of the n integers and checking to see the minimal number of crossings for each, we then have n =&0&1&2&3&4&5&6&7&8&9&10&11\cdots\\ a_0(n) =&1,& 1,& 1,& 2,& 3,& 8,& 14,& 42,& 81,& 262,& 538,& 1828,\cdots\\ a_1(n) =&0,&0,& 1,& 4,& 10,& 36,& 85,& 312,& 737,& 2760,& 6604, &25176,\cdots\\ a_2(n) =&0,&0,& 0,& 0,& 8,& 42,& 168,& 760,& 2418,& 10490,& 30842, &131676,\cdots\\ a_3(n) =&0,&0,& 0,& 0,& 2,& 16,& 164,& 944,& 4386,& 22240,& 83066, &398132,\cdots\\ a_4(n) =&0,&0,& 0,& 0,& 1,& 18,& 146,& 1076,& 6255,& 37250,& 168645, &908898,\cdots\\ a_5(n) =&0,&0,& 0,& 0,& 0,& 0,& 96,& 960,& 7388,& 51968,& 282122, &1711824, \cdots\\ a_6(n) =&0,&0,& 0,& 0,& 0,& 0,& 30,& 440,& 6472,& 55140,& 384065, &2642444,\cdots\\ a_7(n) =&0,&0,& 0,& 0,& 0,& 0,& 14,& 368,& 5176,& 53920,& 455944, &3575040,\cdots\\ a_8(n) =&0,&0,& 0,& 0,& 0,& 0,& 2,& 66,& 3542,& 45960,& 484058, &4336734,\cdots\\ a_9(n) =&0,&0,& 0,& 0,& 0,& 0,& 1,& 72,& 2011,& 32280,& 452504, &4661756,\cdots\\ a_{10}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1172,& 25066,& 396493, &4709856,\cdots\\ a_{11}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 420,& 11840,& 309696, &4291440,\cdots\\ a_{12}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 201,& 8930,& 225754, &3661348,\cdots\\ a_{13}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 40,& 2240,& 151849, &2947392,\cdots\\ a_{14}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 18,& 2040,& 91147, &2103648,\cdots\\ a_{15}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 2,& 224,& 55030, &1575744,\cdots\\ a_{16}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 1,& 270,& 26762, &915924,\cdots\\ a_{17}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 14627, &665088,\cdots\\ a_{18}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 5405, &295956,\cdots\\ a_{19}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2642, &218508,\cdots\\ a_{20}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 641, &63522,\cdots\\ a_{21}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 293, &54672,\cdots\\ a_{22}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 48, &8964,\cdots\\ a_{23}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 22, &9552,\cdots\\ a_{24}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 2, &706,\cdots\\ a_{25}(n) =&0,&0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 0,& 1, &972,\cdots where the vertical sum over columns of terms gives n!. CONJECTURES The above information has lead to a few conjectures. a_{n^2}(2n) = 1 this can be converted to words as, there is exactly one path through 2n points that crosses n² times. The partitions associated with these paths are (2,1)\\ (3,1,4,2)\\ (4,1,5,2,6,3)\\ (5,1,6,2,7,3,8,4)\\ (6,1,7,2,8,3,9,4,10,5) and a clear interlaced pattern can be seen (an example is given in Fig. 3). a_{n^2-1}(2n) = 2, \; n>1 a_{n^2-2}(2n) = 4n+2, \; n>2 a_{n^2-3}(2n) = 8n+8, \; n>3 a_{n^2}(2n+1) = 2(n+1)3^{n-1}, \; n>1

ABSTRACT By introducing a new form of metric tensor the same derivation for the electromanetic tensor Fμν from potentials Aμ leads to the dual space (Hodge Dual) of the regular Fμν tensor. There are additional components in the i, j, k planes, however if after the derivation only the real part is considered a physically consistent electromagnetic theory is recovered with a relabelling of $$ fields to $$ fields and vice versa.

MAIN Let G(q)=_2(m(q)) be an exponential generating function, where Li₂ is the polylogarithm of order 2, _2(z)=^\infty {k^2} and m(q) is the inverse elliptic nome which can be expressed through the Dedakind eta function as m(q)={2})^{8}\eta(2\tau)^{16}}{\eta(\tau)^{24}} where q = eiπτ or by Jacobi theta functions m(q)=\left({\theta_3(0,q)}\right)^4 where \theta_2(0,q)=2^\infty q^{(n+1/2)^2}\\ \theta_3(0,q)=1+2^\infty q^{n^2} giving explicitly G(x)=^\infty {k^2}\left(^\infty x^{(n+1/2)^2}}{1+2^\infty x^{n^2}}\right)^{4k}=^\infty {k!} if we consider the sequence of coefficients ak associated with G(x), modulo 1, or the fractional part of the coefficients, frac(ak) we gain the following sequence 0,0,0,{3},0,{5},0,{7},0,0,0,{11},0,{13},0,0,0,{17},0,{19},0,0,0,{23},0,0,0,0,0,{19},0,{31},0,0,0,0,0,{37},0,0,0,{41},\cdots we see the primes in the denominator in positions where the power of x is a prime. We also note that so far, the numerators are always less than the denominator (obviously), but count, succesively upwards, producing monotonically increasing subsequences. The prime only parts continue {3},{5},{7},{11},{13},{17},{19},{23},{29},{31},{37},{41},{43},{47},{53},{59},{61},{67},{71},{73},{79},{83},{89},{97}, After closer inspection, we see the numerators from the point 1, 3, 7, 13, 15, 21, 25, 27, 31, 37, 43, 45, 51, 55, 57, ... take the form prime(k)−16, the numerators before this take the form 2 ⋅ prime(k)−16, for 6, 10, 3 ⋅ prime(k)−16 for 5, 4 ⋅ prime(k)−16 for 4 and 6 ⋅ prime(k)−16 for the first numerator 2. It is likely then that for the rest of the numbers this pattern continues. This then gives for the coefficient ak of G(x), with k > 6, (a_k)={k}, \; k\in We find that if we take the original coefficients ak, and subtract this fractional part in general \delta_k=a_k-{k} for numbers m which cannot be written as a sum of at least three consecutive positive integers, δm is an integer (empirical). A111774 “ Numbers that can be written as a sum of at least three consecutive positive integers.” apart from odd primes, numbers which cannot are powers of two. OTHER We find a similar relationship with G_2(x)=_2\left({(1-x)^2\left(1-{x-1}\right)^2}\right)=^\infty {k!} where bk seem to follow for k > 2 (b_k)={k}, \; k\in GENERATING FUNCTION FOR FRACTIONAL PART We see the Generating function for n/2 is {2(x-1)^2} but the generating function for the fractional part of n/2, which is (n mod2)/2, is given by {2(x^2-1)} the property described is associated with the polylog, and we seen that the fractional part of _2(2x)=^\infty {k!} gives (c_k)={k}, \; k\in\\ 0, \; this means \left({k^2}\right) = {k}, \; k\in\\ 0,\; or \left({k}\right)= {k}, \; k\in\\ 0,\; we also see that \left({k}\right)= {k}, \; k\in\\ {2}, 4\\ 0,\;

ABSTRACT We develop a triplet operator system which encompasses the structure of quark combinations. Ladder operators are created. The constants β are currently being found.