Conversion from partial fraction expansion to continued fraction

The partial fraction expansions of tangent and cotangent are

    in regular form

       in regular form

That is

To convert these, we usually use the following formula

  (regular expression in series)

   in special form

or

Applying gamma-shape multipliers, we get

  in special form

Thus

  in special form

where 5=1²+2², 13=2²+3², 25=3²+4² after third term in order.

In the same way

or

Applying gamma-shape multipliers, we get

Thus

in special form

where 10=1²+3², 36=3²+5², 74=5²+7², 130=7²+9² after second term in order.