List manipulation - various

Question

Update

OK, Please forgive the messiness of this, but I am working with something like:

d = (q = 8;
f = (lop = 
  Transpose[{Flatten[
     Reverse[Table[
        Gamma[y], {y, 1, q + 1}]] /. {ComplexInfinity -> 0}], 
    Flatten[Table[Binomial[n, k], {n, q, q}, {k, 0, n}]], 
    Flatten[Table[x^n, {n, 0, q}]]}];
 {#1*#2*#3} & @@@ lop);
Flatten[f];
Total[f]);
d

which gives:

{40320 + 40320 x + 20160 x^2 + 6720 x^3 + 1680 x^4 + 336 x^5 + 56 x^6 + 8 x^7 + x^8}

I am then comparing the roots with another series, eg:

Normal[Series[E^x, {x, 0, 8}]]

which, in this case are the same.

However, I have lots of different series & was wondering if I could manipulate them as stated below.

(NB - I realise there is no 'ComplexInfinity' output in the above example, but have left it in to demonstrate that I have tried several replace techniques for the subsequent questions.)

Original question

I have several questions on list manipulation & at the risk of asking the same question multiple times, I have compiled them into the following single question:

(a) Changing alternate (or some other pattern) signs / operators:

From this:

1 + x + (x^2)/2! +  (x^3)/3! + (x^4)/4! +  (x^5)/5! ...

I would like to do get this:

1 - x + (x^2)/2! -  (x^3)/3! + (x^4)/4! -  (x^5)/5! ...

(b) Multiplying alternate (or some other pattern) elements by a given value (eg - I):

From this:

1 + x + (x^2)/2! +  (x^3)/3! + (x^4)/4! +  (x^5)/5! ...

I would like to do get this:

1 - I x + (x^2)/2! -  I(x^3)/3! + (x^4)/4! -  I(x^5)/5! ...

(c) Convert all Imaginary values in a list to real values (or vice versa), noting that there may be no pattern in the distribution here:

From this:

1 + I x + (x^2)/2! +  (x^3)/3! + I(x^4)/4! +  I(x^5)/5! ...

I would like to do get this:

1 + x + (x^2)/2! +  (x^3)/3! + (x^4)/4! +  (x^5)/5! ...

or this:

I + I x + I(x^2)/2! +  I(x^3)/3! + I (x^4)/4! +  I(x^5)/5! ...

I have tried various select & replace methods, but have had no luck so far.

Answer 1

Well you can begin by wrapping your expression in Hold or HoldForm then using ReleaseHold when you want to Evaluate them. For your first example:

expr = HoldForm[1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + (x^5)/5!]

Now you can do:

Replace[expr /. {Times[c : Power[x, n_?OddQ], d_] :> Times[-1, c, d]}, x -> - x, 2]

To get:

1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5!

For the next one:

Replace[expr /. {Times[c : Power[x, n_?OddQ], d_] :> Times[-I, c, d]}, x -> - I x, 2]

Which gives:

1 - I x + x^2/2! - (I x^3)/3! + x^4/4! - (I x^5)/5!

For the third:

expr2 = 1 + I x + (x^2)/2! + (x^3)/3! + I (x^4)/4! + I (x^5)/5!;

To change the imaginary numbers do:

expr2 /. Complex[0, c_] :> c

To change negatives to positives in e.g.

expr3 = -1 - x + (x^2)/2! - (x^3)/3! + (x^4)/4! - (x^5)/5!

Simply do:

expr3 /. {Times[-1, c_] :> Times[1, c], Rational[-1, d_] :> Rational[1, d], n_?Negative :> -1 n}

To get:

1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120