Issue No. 06 - June (1972 vol. 21)
H. Butin , Telecommunications Division, Thomson-CSF, Gennevilliers, France.
A polynomial definition of the Walsh functions (very similar to that used by Titsworth  or Pearl ) is shown to be equivalent to Pichler-Harmuth's formulas . It gives an easy way to derive some properties of Walsh functions, such as their relationship with the Gray code or the shifting theorem.
H. Butin, "A Compact Definition of Walsh Functions", IEEE Transactions on Computers, vol. 21, no. , pp. 590-592, June 1972, doi:10.1109/TC.1972.5009014