ABSTRACT We investigate an apparently fundamental operation on commonly occurring mathematical series. MAIN If we take the hypergeometric series for example: \;_2F_1(a,b,c,x) = ^\infty {(c)_k}{k!} we can notice that for the transform $$ _n[f(n)](m) = ^m f(n) = g(m) $$ which is almost the indefinite product, we have the inverse transform of the summation kernel $$ ^{-1}_k \left[{(c)_k}{k!}\right](n) = {(n+c-1)}{n} $$ we now proceed to define a generating function of this new kernel as G(t) = ^\infty {(n+c-1)}{n} t^n with $$ G(t) = _1(2,c;c+1;t))}{c-1} $$ for an example of $f(x) = {\pi}K(x)$ with a = b = 1/2, c = 1, then $$ G(t) = _2(t)}{4}-{t-1}+x \log (1-t) $$ we have that the inverse Z-transform of $G({t})$ gives $$ ^{-1}\left[G({t})\right] = {4n^2}, n>0 $$ and we can re-extrude this as $$ \prod_n {4n^2} = {2}+n)x^n}{\pi \Gamma(1 + n)^2} $$ yielding $$ ^\infty ^n {4n^2} = {\pi}K(x) $$ TRANSFORM From this we essentially have an operator $_x$ that maps a function to the discrete difference reduced term, for the hypergeometric example this would be $$ _x[\,_2F_1(a,b,c,x)] = {(n+c-1)}{n}. $$ The inverse operation $^{-1}$ is then exactly $$ ^{-1}_n[\square](x) = ^\infty ^m x \square $$ this is used as $$ ^{-1}_n[f(n)](x) = ^\infty ^m x f(n) $$ this means the operator Q is a composition of “coefficient of” operator, commonly denoted [xn] and the discrete difference derivative Δn* as $$ _x[\square](n) = \Delta^*_n [x^n] \square $$ CONNECTION TO MELLIN TRANSFORM We can connect this to the Mellin transform and the Ramanujan master theorem, which essentially extracts coefficients. For a function f(x) = ^\infty {k!} \phi(k) x^k we have that the Mellin transform is related to the coefficient function by $$ [f](s) = \Gamma(s)\phi(-s) $$ for suitable functions. In effect this becomes the method of coefficient extraction, but brings a sign flip, σ, operation in. Thus for a function defined as in equation [eqn:RMT] we have $$ Q f = \Delta^* \sigma {\Gamma(s)} ^{-1} f $$ then an operator G would indicate summing over positive non-zero integers $$ G_n[\square](t) = ^\infty t^n \square $$ some important identities that are not immediately obvious when reducing more complex series expansions such as elliptic integrals ^n {(2n-2)!} = \Gamma(2n+1) \\ ^n {(mn-m)!} = \Gamma(mn+1) \\ ^n {(mn-m+b)!} = {b!} \\ ^n {1-2n} = {1-2n} EXAMPLES Transforms from function to generating function: G Q[e^x] = -\log(1-t) \\ G Q[e^{-x}] = \log(1-t) \\ GQ\left[{1-x}\right] = {1-t} \\ GQ\left[{1+x}\right] = -{1-t} \\ GQ\left[I_0()\right] = _2(t)}{4} \\ GQ\left[I_0()\right] = _2(t)}{4} \\ GQ\left[J_0()\right] = -_2(t)}{4} \\ GQ[ {\pi} K(x)] = _2(t)}{4}-{t-1}+\log (1-t) \\ GQ[ {\pi} E(x)] = _2(t)}{4}+{1-t}+2\log (1-t) \\ GQ[ )}{}] = 3 + {1-t} - 4 {}}{} - {2}\log(1-t) \\ GQ[ 3)/3)}{}] = -{t-1}-{9} \log (1-t)-\left(\right)}{9 }+{9} \\ GQ\left[(1-x)^{-5/9}\right] = {1-t} + {9} \log(1-t) \\ GQ\left[(1-x)^{a-1}\right] = {1-t} + a \log(1-t)\\ GQ\left[1-\tanh^{-1}()\right] = {1-t} - 2 \tanh^{-1}() \\ \cosh() \to \tanh^{-1}() + {2}\log(1-t) \\ \cos() \to - \tanh^{-1}() - {2}\log(1-t) \\ ^\infty {(3k)!} \to {2} \, _2F_1\left({3},1;{3};t\right)-{2} \, _2F_1\left({3},1;{3};t\right)-{6} \log (1-t) here we see that $$ GQ\left[ {\pi} K(x)\right] = GQ\left[I_0()\right] + GQ\left[{1+x}\right] + G Q[e^{-x}] $$ There could be some secret equivalence between the function on the left and that on the right. I.e. the elliptic K function may transform under an operator and the combination of functions on the right may transform in analogy under a different operator. For example $$ tD_t \pm \log(1-t) \to {1-t} $$ so this meta derivative converts $$ e^{\pm x} \to {1\pm x} $$ this can be seen to be similar to an inverse Borel transform! From the above list of transforms it is clear that we see repeating units or “elements”, for example log(1 − t) is very common. It may be instructive to find a naming system for these units to give a compact representation of the resulting function. Whether these elements form some kind of basis for the underlying function space is yet to be investigated. We appear to have functions of the form x ₂F₁(a, b, c, x), or at least for shorthand -\log(1-t) = t\;_2F_1(1,1,2,t) = t_{1,1;2} \\ \sin^{-1}() = t_{{2},{2};{2}} with this we can immediately see $$ ^\infty {(3k)!} \to {3},1;{3}}}{2}-{3},1;{3}}}{2}+}{6} = {2} & -{2} & {6} t_{{3},1;{3}} \\ t_{{3},1;{3}} \\ t_{1,1;2} $$ important terms might include $$ ^\infty H_n t^n = -{1-t} = }{1-t} $$ to handle this we would need to evaluate $$ ^n H_k = f(n) $$ and apparently little is understood about these terms in OEIS A097423 and A097424. DERIVATIVES Consider the derivative of a sequence, we have $$ {dx} ^\infty a_k x^k = ^\infty (k+1)a_{k+1}x^k $$ where we have made sure to keep the sequence index from 0 to ∞. We can write $$ ^k {n} = k+1 $$ which tells us $$ Q[f'(x)] = ^\infty {n} \Delta^*_k[a_{k+1}](n) t^n $$ for example if for ex we have ak = 1/k!, then the derivative gives $$ Q[e^x] = ^\infty {n} \Delta^*_k[{k!}](n) t^n $$ and $\Delta^*_k[{k!}](n) = (n+1)^{-1}$ which consistently gives $$ Q[e^x] = ^\infty {n} t^n = -\log(1-t) $$ this is powerful, and we can use this to calculate unknown derivatives Δk*, and potentially solve differential equations in a mirror domain. In general we have a beautiful relationship $$ {dx^n} ^\infty a_k x^k = ^\infty (k+1)_n a_{k+n}x^k $$ for ex this means $$ \Delta^*_k[{\Gamma(k+n+1)}]= \Delta^*_k[{\Gamma(k+1)}] = {n} $$
